SOLVING PARABOLIC STOCHASTIC PARTIAL DIFFERENTIAL

MATHEMATICS OF COMPUTATION Volume 78, Number 268, October 2009, Pages 2075–2106 S 0025-5718(09)02250-9 Article electronically published on March 6, 2009

SOLVING PARABOLIC STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS VIA AVERAGING OVER CHARACTERISTICS G. N. MILSTEIN AND M. V. TRETYAKOV Abstract. The method of characteristics (the averaging over the characteristic formula) and the weak-sense numerical integration of ordinary stochastic differential equations together with the Monte Carlo technique are used to propose numerical methods for linear stochastic partial differential equations (SPDEs). Their orders of convergence in the mean-square sense and in the sense of almost sure convergence are obtained. A variance reduction technique for the Monte Carlo procedures is considered. Layer methods for linear and semilinear SPDEs are constructed and the corresponding convergence theorems are proved. The approach developed is supported by numerical experiments.

1. Introduction A lot of attention has recently been paid to numerical methods for stochastic partial differential equations (SPDEs). Various numerical approaches for linear SPDEs are considered, e.g., in [31, 1, 19, 17, 34, 10, 11, 3, 7, 25, 5] (see also the references therein). The interest in linear SPDEs of parabolic type is mainly due to their wellknown relation with the nonlinear filtering problem [18, 16, 28, 32, 31] although they have other applications as well (see, e.g., [32] and the references therein). At the same time, very little has been done in studying applications of the Monte Carlo technique to solving SPDEs while such a technique is the well-established tool for solving problems of mathematical physics associated with multi-dimensional (deterministic) partial differential equations (see, e.g., [23] and the references therein). The aim of this paper is to exploit the method of characteristics (the averaging over the characteristic formula, the generalization of the Feynman-Kac formula) and numerical integration of (ordinary) stochastic differential equations (SDEs) together with the Monte Carlo technique to propose numerical methods for linear SPDEs of parabolic type. The Monte Carlo methods are efficient for solving highdimensional SPDEs when functionals or individual values of the solution have to be found. We note that the method of characteristics was exploited in [3] to propose a particle method for the Kushner-Stratonovich equation, and it was used in [25] for constructing Monte Carlo methods for a less general class of SPDEs than the one considered here. Received by the editor May 30, 2007 and, in revised form, November 3, 2008. 2000 Mathematics Subject Classification. Primary 65C30, 60H15, 60H35, 60G35. Key words and phrases. Probabilistic representations of solutions of stochastic partial differential equations, numerical integration of stochastic differential equations, Monte Carlo technique, mean-square and almost sure convergence, layer methods. c ⃝2009 American Mathematical Society

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The probabilistic approach based on the method of characteristics is also exploited to construct new layer methods for SPDEs (see this idea in the case of deterministic PDEs in [21, 23]). The layer methods are competitive with finite difference schemes (see [10, 34] and the references therein); they can be used when one needs to find the SPDE solution everywhere in the space-time domain [𝑡0 , 𝑇 ] × R𝑑 . It was shown in the deterministic case [21, 23] that layer methods have some advantages in comparison with conventional PDE solvers. We expect that they possess remarkable properties in the SPDE case as well. We construct layer methods both for linear and semilinear SPDEs. Semilinear SPDEs are used for modelling in physics, biology and chemistry (see [4, 13] and the references therein). For other numerical approaches to semilinear SPDEs, see, e.g., in [8, 12, 2] (see also the references therein). In Section 2 we recall probabilistic representations (the method of characteristics) for SPDEs of parabolic type from [15, 16, 28, 32]. In Section 3 we propose a number of approximation methods for the SPDEs and study their mean-square and almost sure (a.s.) convergence. The methods are based on approximate solving of the characteristic SDEs, for which we exploit both the mean-square and weaksense numerical integration. Section 4 deals with variance reduction methods that are important for any Monte Carlo procedures. We propose layer methods for linear SPDEs in Section 5 and for semilinear SPDEs in Section 6. Some results of numerical experiments are presented in Section 7. We solve numerically the Ornstein-Uhlenbeck SPDE and, in particular, demonstrate the effectiveness of the proposed variance reduction technique. 2. Conditional probabilistic representations of solutions to linear SPDEs Let (Ω, ℱ, 𝑃 ) be a probability space, ℱ𝑡 , 𝑇0 ≤ 𝑡 ≤ 𝑇, be a nondecreasing family of 𝜎-subalgebras of ℱ, (𝑤(𝑡), ℱ𝑡 ) = ((𝑤1 (𝑡), . . . , 𝑤𝑞 (𝑡))⊤ , ℱ𝑡 ) be a 𝑞-dimensional standard Wiener process. Consider the Cauchy problem for the backward SPDE (2.1) 𝑞 ∑ −𝑑𝑣 = [ℒ𝑣 + 𝑓 (𝑡, 𝑥)] 𝑑𝑡 + [ℳ𝑟 𝑣 + 𝛾𝑟 (𝑡, 𝑥)] ∗ 𝑑𝑤𝑟 (𝑡), (𝑡, 𝑥) ∈ [𝑇0 , 𝑇 ) × R𝑑 , 𝑟=1

𝑣(𝑇, 𝑥) = 𝜑(𝑥), 𝑥 ∈ R𝑑 ,

(2.2) where (2.3) ℒ𝑣(𝑡, 𝑥) := ℳ𝑟 𝑣(𝑡, 𝑥) :=

𝑑 𝑑 ∑ ∂2 ∂ 1 ∑ 𝑖𝑗 𝑎 (𝑡, 𝑥) 𝑖 𝑗 𝑣(𝑡, 𝑥) + 𝑏𝑖 (𝑡, 𝑥) 𝑖 𝑣(𝑡, 𝑥) + 𝑐(𝑡, 𝑥)𝑣(𝑡, 𝑥), 2 𝑖,𝑗=1 ∂𝑥 ∂𝑥 ∂𝑥 𝑖=1 𝑑 ∑ 𝑖=1

𝛼𝑟𝑖 (𝑡, 𝑥)

∂ 𝑣(𝑡, 𝑥) + 𝛽𝑟 (𝑡, 𝑥)𝑣(𝑡, 𝑥), 𝑟 = 1, . . . , 𝑞. ∂𝑥𝑖

The notation “∗𝑑𝑤𝑟 ” in (2.1) means the backward Ito integral. We recall [32] that to define this integral one introduces the “backward” Wiener processes (2.4)

𝑤 ˜𝑟 (𝑡) := 𝑤𝑟 (𝑇 ) − 𝑤𝑟 (𝑇 − (𝑡 − 𝑇0 )), 𝑟 = 1, . . . , 𝑞, 𝑇0 ≤ 𝑡 ≤ 𝑇,

and a decreasing family of 𝜎-subalgebras ℱ𝑇𝑡 , 𝑇0 ≤ 𝑡 ≤ 𝑇, induced by the increments ˜𝑟 (𝑡′ ), 𝑡′ ≤ 𝑡, coincides 𝑤𝑟 (𝑇 ) − 𝑤𝑟 (𝑡′ ), 𝑟 = 1, . . . , 𝑞, 𝑡′ ≥ 𝑡. A 𝜎-algebra induced by 𝑤

SOLVING PARABOLIC STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

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𝑇 −(𝑡−𝑇 )

0 with ℱ𝑇 . Then the backward Ito integral is defined as the Ito integral with respect to 𝑤(𝑠): ˜ ∫ 𝑇 −(𝑡−𝑇0 ) ∫ 𝑡′ ′′ ′′ 𝜓(𝑡 ) ∗ 𝑑𝑤𝑟 (𝑡 ) := 𝜓(𝑇 − (𝑡′′ − 𝑇0 ))𝑑𝑤 ˜𝑟 (𝑡′′ ), 𝑇0 ≤ 𝑡 ≤ 𝑡′ ≤ 𝑇,

𝑡

𝑇 −(𝑡′ −𝑇0 )

𝑇 −(𝑡−𝑇 )

0 -adapted square-integrable function. where 𝜓(𝑇 − (𝑡 − 𝑇0 )), 𝑡 ≤ 𝑇, is an ℱ𝑇 𝑡 The process 𝑣(𝑡, 𝑥) is ℱ𝑇 -adapted; it depends on 𝑤(𝑠) − 𝑤(𝑡), 𝑡 ≤ 𝑠 ≤ 𝑇. We pay attention that the more precise notation for the solution of (2.1)-(2.2) is 𝑣(𝑡, 𝑥; 𝜔), 𝜔 ∈ Ω, but we use, as a rule, the shorter one, 𝑣(𝑡, 𝑥).

Assumption 2.1. We assume that the coefficients in (2.1) are sufficiently smooth and that their derivatives up to some order are bounded (in particular, it follows from here that the coefficients are globally Lipschitz). Furthermore, it is supposed that 𝑐, 𝛽𝑟 , 𝑓, and 𝛾𝑟 are bounded themselves. Assumption 2.2. We assume that the function 𝜑(𝑥) is also sufficiently smooth and that 𝜑(𝑥) and its derivatives up to some order belong to the class functions satisfying an inequality of the form (2.5)

∣𝜑(𝑥)∣ ≤ 𝐾(1 + ∣𝑥∣ϰ ), 𝑥 ∈ R𝑑 ,

where 𝐾 and ϰ are positive constants. Assumption 2.3. We assume that 𝑎 = {𝑎𝑖𝑗 } is symmetric and that the matrix 𝑎 − 𝛼𝛼⊤ is nonnegative definite (the coercivity condition). Assumptions 2.1-2.3 ensure the existence of a unique classical solution 𝑣(𝑡, 𝑥) of (2.1)-(2.2) which has derivatives in 𝑥𝑖 , 𝑖 = 1, . . . , 𝑑, up to a sufficiently high order satisfying an inequality of the form (2.5) a.s. with a positive random variable 𝐾 = 𝐾(𝜔) for which the moments of a sufficiently high order are bounded (see [14, 32]). They are sufficient for all the statements in Sections 3 and 4 (some additional assumptions are needed in Section 5). At the same time, they are not necessary and the methods constructed can be used under broader conditions. We note that an attempt to weaken the conditions would inevitably lead to difficulties of a technical nature and, as a result, to a less clear exposition together with an unnecessary increase of the paper’s length. Let a 𝑑 × 𝑝 matrix 𝜎(𝑡, 𝑥) be obtained from the equation 𝜎(𝑡, 𝑥)𝜎 ⊤ (𝑡, 𝑥) = 𝑎(𝑡, 𝑥) − 𝛼(𝑡, 𝑥)𝛼⊤ (𝑡, 𝑥). The solution of the problem (2.1)-(2.2) has the following conditional probabilistic representation (the conditional Feynman-Kac formula) [15, 16, 28, 32]: (2.6)

𝑣(𝑡, 𝑥) = 𝐸 𝑤 [𝜑(𝑋𝑡,𝑥 (𝑇 ))𝑌𝑡,𝑥,1 (𝑇 ) + 𝑍𝑡,𝑥,1,0 (𝑇 )] , 𝑇0 ≤ 𝑡 ≤ 𝑇,

where 𝑋𝑡,𝑥 (𝑠), 𝑌𝑡,𝑥,𝑦 (𝑠), 𝑍𝑡,𝑥,𝑦,𝑧 (𝑠), 𝑡 ≤ 𝑠 ≤ 𝑇, is the solution of the SDEs [ ] 𝑞 ∑ 𝑑𝑋 = 𝑏(𝑠, 𝑋) − 𝛼𝑟 (𝑠, 𝑋)𝛽𝑟 (𝑠, 𝑋) 𝑑𝑠 (2.7) +

𝑝 ∑

𝑟=1

𝜎𝑟 (𝑠, 𝑋)𝑑𝑊𝑟 (𝑠) +

𝑟=1

𝑞 ∑ 𝑟=1

𝑋(𝑡) = 𝑥,

𝛼𝑟 (𝑠, 𝑋)𝑑𝑤𝑟 (𝑠),

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𝑑𝑌 = 𝑐(𝑠, 𝑋)𝑌 𝑑𝑠 + 𝑑𝑍 = 𝑓 (𝑠, 𝑋)𝑌 𝑑𝑠 +

𝑞 ∑ 𝑟=1 𝑞 ∑

𝛽𝑟 (𝑠, 𝑋)𝑌 𝑑𝑤𝑟 (𝑠), 𝑌 (𝑡) = 𝑦, 𝛾𝑟 (𝑠, 𝑋)𝑌 𝑑𝑤𝑟 (𝑠), 𝑍(𝑡) = 𝑧,

𝑟=1

and 𝑊 (𝑠) = (𝑊1 (𝑠), . . . , 𝑊𝑝 (𝑠))⊤ is a 𝑝-dimensional standard Wiener process independent of 𝑤(𝑠), and the expectation 𝐸 𝑤 in (2.6) is taken over the realizations of 𝑊 (𝑠), 𝑡 ≤ 𝑠 ≤ 𝑇, for a fixed 𝑤(𝑠), 𝑡 ≤ 𝑠 ≤ 𝑇 ; in other words, 𝐸 𝑤 (⋅) means the conditional expectation 𝐸 (⋅∣𝑤(𝑠) − 𝑤(𝑡), 𝑡 ≤ 𝑠 ≤ 𝑇 ) . Remark 2.1. Introduce the new time variable 𝑠 := 𝑇 − (𝑡 − 𝑇0 ) and introduce the ˜𝑖𝑗 (𝑠, 𝑥) := 𝑎𝑖𝑗 (𝑇 + 𝑇0 − 𝑠, 𝑥), ˜𝑏𝑖 (𝑠, 𝑥) := functions 𝑢(𝑠, 𝑥) := 𝑣(𝑇 + 𝑇0 − 𝑠, 𝑥), 𝑎 𝑖 ˜ 𝑟𝑖 (𝑠, 𝑥), 𝛽˜𝑟 (𝑠, 𝑥), 𝛾˜𝑟 (𝑠, 𝑥). Then 𝑏 (𝑇 + 𝑇0 − 𝑠, 𝑥), and analogously 𝑐˜(𝑠, 𝑥), 𝑓˜(𝑠, 𝑥), 𝛼 one can show that 𝑢(𝑠, 𝑥) is the solution of the Cauchy problem for the forward SPDE (see [32, p. 173]): ⎤ ⎡ 𝑑 𝑑 2 ∑ ∑ 1 ∂ ∂ ˜𝑏𝑖 (𝑠, 𝑥) (2.8) 𝑎 ˜𝑖𝑗 (𝑠, 𝑥) 𝑖 𝑗 𝑢 + 𝑢 + 𝑓˜(𝑠, 𝑥)⎦ 𝑑𝑠 𝑑𝑢 = ⎣ 𝑖 2 𝑖,𝑗=1 ∂𝑥 ∂𝑥 ∂𝑥 𝑖=1 ] [ 𝑑 𝑞 ∑ ∑ ∂ 𝑖 + 𝛼 ˜ 𝑟 (𝑠, 𝑥) 𝑖 𝑢 + 𝛽˜𝑟 (𝑠, 𝑥)𝑢 + 𝛾˜𝑟 (𝑠, 𝑥) 𝑑𝑤 ˜𝑟 (𝑠), (𝑠, 𝑥) ∈ (𝑇0 , 𝑇 ] × R𝑑 , ∂𝑥 𝑟=1 𝑖=1 𝑢(𝑇0 , 𝑥) = 𝜑(𝑥), 𝑥 ∈ R𝑑 ,

(2.9)

where 𝑤 ˜𝑟 (𝑠) are defined in (2.4). The process 𝑢(𝑠, 𝑥) is ℱ𝑇𝑇 +𝑇0 −𝑡 -adapted; it depends on 𝑤 ˜𝑟 (𝑠′ ), 𝑟 = 1, . . . , 𝑞, 𝑇0 ≤ 𝑠′ ≤ 𝑠. Analogously, for a given forward SPDE one can write down the corresponding backward SPDE. Thus, the methods for backward SPDEs considered in this paper can be used for solving forward SPDEs as well. Remark 2.2. Consider an infinite-dimensional Wiener process 𝐵(𝑡) taking values in some Hilbert space 𝐻 and with covariance operator 𝑄 (which is assumed to be a nuclear operator). Let 𝑒𝑟 be unit eigenvectors of 𝑄 with nonzero √ eigenvalues 𝜆𝑟 = (𝑄𝑒𝑟 , 𝑒𝑟 ). Then (see, e.g. [4, 14, 32]) 𝑤𝑟 (𝑡) := (𝑒𝑟 , 𝐵(𝑡))/ 𝜆𝑟 are independent standard Wiener processes, and the infinite-dimensional Wiener process is represented as ∞ √ ∑ 𝜆𝑟 𝑤𝑟 (𝑡)𝑒𝑟 . (2.10) 𝐵(𝑡) = 𝑟=1

Furthermore, for any 𝐻-valued process 𝜓(𝑠) for which the integral defined, one has [14, 32] ∫ 𝑡 ∞ ∫ 𝑡 ∑ (2.11) 𝜓(𝑠)𝑑𝐵(𝑠) = 𝜓𝑟 (𝑠)𝑑𝑤𝑟 (𝑡) 0

𝑟=1

0

∫𝑡 0

𝜓(𝑠)𝑑𝐵(𝑠) is

∫𝑡 √ with 𝜓𝑟 (𝑠) = (𝜓(𝑠), 𝑒𝑟 ) 𝜆𝑟 . The Wiener process 𝐵(𝑡) and the integral 0 𝜓(𝑠)𝑑𝐵(𝑠) can be approximated by truncating the expansions in (2.10) and (2.11) (see, e.g., [4]). Then it is possible to view the SPDEs (2.1) and (2.8) as approximations of SPDEs driven by the infinite-dimensional Wiener process. Consequently, the methods proposed in this paper for (2.1) can, in principle, be used for approximating


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SPDEs with infinite-dimensional Wiener process, but we do not consider this aspect any further here. 3. Numerical methods based on the conditional Feynman-Kac formula (2.6)-(2.7) Our purpose is to simulate approximately the random variable 𝑣(𝑡, 𝑥, 𝜔) under any fixed (𝑡, 𝑥) using the probabilistic representation (2.6)-(2.7). In this section we construct mean-square approximations 𝑣¯(𝑡, 𝑥, 𝜔) (see Section 3.1) and 𝑣˜(𝑡, 𝑥, 𝜔) (see Section 3.2) for 𝑣(𝑡, 𝑥, 𝜔); i.e., we construct 𝑣¯(𝑡, 𝑥, 𝜔) (or 𝑣˜(𝑡, 𝑥, 𝜔)) to be close to 𝑣(𝑡, 𝑥, 𝜔) in the mean-square sense. For a realization of the probabilistic representation (2.6), one can use numerical integration of SDEs (2.7) with respect to the Wiener process 𝑊 (𝑠) both in the mean-square sense (Section 3.1; the approximation 𝑣¯(𝑡, 𝑥)) and in the weak sense (Section 3.2; the approximation 𝑣˜(𝑡, 𝑥)). We prove a.s. convergence of the proposed methods. 3.1. Mean-square simulation of the representation (2.6)-(2.7). Consider the one-step approximation for (2.7): [ ] 𝑞 ∑ ¯ 𝑡,𝑥 (𝑡 + ℎ) = 𝑥 + ℎ 𝑏(𝑡, 𝑥) − 𝛼𝑟 (𝑡, 𝑥)𝛽𝑟 (𝑡, 𝑥) 𝑋 (3.1) +

𝑞 ∑

𝑟=1

𝛼𝑟 (𝑡, 𝑥)Δ𝑡 𝑤𝑟 +

𝑟=1

𝑌¯𝑡,𝑥,𝑦 (𝑡 + ℎ) = 𝑦 + ℎ𝑐(𝑡, 𝑥)𝑦 +

𝑞 ∑

𝑝 ∑

𝜎𝑟 (𝑡, 𝑥)Δ𝑡 𝑊𝑟 ,

𝑟=1

𝛽𝑟 (𝑡, 𝑥)𝑦Δ𝑡 𝑤𝑟 ,

𝑟=1 𝑞 ∑

𝑍¯𝑡,𝑥,𝑦,𝑧 (𝑡 + ℎ) = 𝑧 + ℎ𝑓 (𝑡, 𝑥)𝑦 +

𝛾𝑟 (𝑡, 𝑥)𝑦Δ𝑡 𝑤𝑟 ,

𝑟=1

where Δ𝑡 𝑤𝑟 := 𝑤𝑟 (𝑡 + ℎ) − 𝑤𝑟 (𝑡), Δ𝑡 𝑊𝑟 := 𝑊𝑟 (𝑡 + ℎ) − 𝑊𝑟 (𝑡). The approximation (3.1) generates the strong Euler method in the usual way. We introduce a partition of the time interval [𝑡, 𝑇 ], for simplicity the equidistant one: 𝑡 = 𝑡0 < ⋅ ⋅ ⋅ < 𝑡𝑁 = 𝑇, with step size ℎ = (𝑇 − 𝑡)/𝑁. An approximation of ¯ 𝑘 , 𝑌¯𝑘 , 𝑍¯𝑘 ). (𝑋𝑡,𝑥 (𝑡𝑘 ), 𝑌𝑡,𝑥,1 (𝑡𝑘 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘 )) is denoted by (𝑋 The Euler scheme takes the form (3.2) [ ] 𝑞 ∑ ¯ 0 = 𝑥, 𝑋 ¯ 𝑘+1 = 𝑋 ¯ 𝑘 + ℎ 𝑏(𝑡𝑘 , 𝑋 ¯𝑘 ) − ¯ 𝑘 )𝛽𝑟 (𝑡𝑘 , 𝑋 ¯𝑘 ) 𝑋 𝛼𝑟 (𝑡𝑘 , 𝑋 +

𝑞 ∑ 𝑟=1

¯ 𝑘 )Δ𝑘 𝑤𝑟 + 𝛼𝑟 (𝑡𝑘 , 𝑋

𝑝 ∑

𝑟=1

¯ 𝑘 )Δ𝑘 𝑊𝑟 , 𝜎𝑟 (𝑡𝑘 , 𝑋

𝑟=1

¯ 𝑘 )𝑌¯𝑘 + 𝑌¯0 = 1, 𝑌¯𝑘+1 = 𝑌¯𝑘 + ℎ𝑐(𝑡𝑘 , 𝑋

𝑞 ∑

¯ 𝑘 )𝑌¯𝑘 Δ𝑘 𝑤𝑟 , 𝛽𝑟 (𝑡𝑘 , 𝑋

𝑟=1 𝑞 ∑

¯ 𝑘 )𝑌¯𝑘 + 𝑍¯0 = 0, 𝑍¯𝑘+1 = 𝑍¯𝑘 + ℎ𝑓 (𝑡𝑘 , 𝑋

¯ 𝑘 )𝑌¯𝑘 Δ𝑘 𝑤𝑟 , 𝑘 = 0, . . . , 𝑁 − 1, 𝛾𝑟 (𝑡𝑘 , 𝑋

𝑟=1

where Δ𝑘 𝑤𝑟 := 𝑤𝑟 (𝑡𝑘+1 ) − 𝑤𝑟 (𝑡𝑘 ), Δ𝑘 𝑊𝑟 := 𝑊𝑟 (𝑡𝑘+1 ) − 𝑊𝑟 (𝑡𝑘 ).

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Let (3.3)


[ ] ¯ 𝑁 )𝑌¯𝑁 + 𝑍¯𝑁 , 𝑣¯(𝑡, 𝑥) := 𝐸 𝑤 𝜑(𝑋

¯ 𝑁 , 𝑌¯𝑁 , 𝑍¯𝑁 are from (3.2). where 𝑋 Below we use the same letter 𝐶 = 𝐶(𝑥) or 𝐶 = 𝐶(𝑥, 𝜔) for various expressions of the form 𝐾(1 + ∣𝑥∣ϰ ) (see (2.5)) with 𝐾 being a positive constant or 𝐾 = 𝐾(𝜔) being a positive random variable, respectively. Theorem 3.1. Let Assumptions 2.1-2.3 hold. The method (3.2)-(3.3) satisfies the inequality for 𝑝 ≥ 1 : ( )1/(2𝑝) 2𝑝 ≤ 𝐶(𝑥)ℎ1/2 , (3.4) 𝐸 ∣¯ 𝑣 (𝑡, 𝑥) − 𝑣(𝑡, 𝑥)∣ where 𝐶 does not depend on the discretization step ℎ, i.e., in particular, (3.2)-(3.3) is of mean-square order 1/2. For almost every trajectory 𝑤(⋅) and any 𝜀 > 0 there exists 𝐶(𝑥, 𝜔) > 0 such that (3.5)

∣¯ 𝑣 (𝑡, 𝑥) − 𝑣(𝑡, 𝑥)∣ ≤ 𝐶(𝑥, 𝜔)ℎ1/2−𝜀 ,

where 𝐶 does not depend on the discretization step ℎ, i.e., the method (3.2)-(3.3) converges with order 1/2 − 𝜀 a.s. ¯ 𝑘 , 𝑌¯𝑘 , 𝑍¯𝑘 )⊤ , ¯ 𝑘 := (𝑋 Proof. Introduce x = (𝑥, 𝑦, 𝑧), X(𝑠) := (𝑋(𝑠), 𝑌 (𝑠), 𝑍(𝑠))⊤ , X and 𝜓(x) := 𝜑(𝑥)𝑦 + 𝑧. ¯ 𝑘 (see [23]) have bounded moments of any It is known that X(𝑠) (see [6]) and X order and also that (see [9]) for 𝑝 ≥ 1, ( )1/(2𝑝) ¯ 𝑘 − X(𝑡𝑘 )2𝑝 ≤ 𝐶(𝑥)ℎ1/2 , (3.6) 𝐸 X where 𝐶 does not depend on ℎ. By the conditional version of Jensen’s inequality, we get ¯ 𝑁 ) − 𝐸 𝑤 𝜓(X(𝑡𝑁 ))2𝑝 𝐸 ∣¯ 𝑣 (𝑡, 𝑥) − 𝑣(𝑡, 𝑥)∣2𝑝 = 𝐸 𝐸 𝑤 𝜓(X ] [ ¯ 𝑁 ) − 𝜓(X(𝑡𝑁 )) 2𝑝 ≤ 𝐸 𝐸 𝑤 𝜓(X [ ] ¯ 𝑁 ) − 𝜓(X(𝑡𝑁 ))2𝑝 ≤ 𝐸 𝐸 𝑤 𝜓(X ¯ 𝑁 ) − 𝜓(X(𝑡𝑁 ))2𝑝 . = 𝐸 𝜓(X Using the smoothness of 𝜓 and the assumption that derivatives of 𝜓 satisfy an inequality of the form (2.5), we obtain 𝜓(X ¯ 𝑁 ) − 𝜓(X(𝑡𝑁 )) ≤ 𝐾(1 + ∣X(𝑡𝑁 )∣ϰ + ∣X ¯ 𝑁 ∣ϰ ) X ¯ 𝑁 − X(𝑡𝑁 ) , where 𝐾 is a positive constant. Then due to the boundedness of the moments, the Cauchy-Bunyakovskii inequality, and (3.6), we get ¯ 𝑁 ∣ϰ )2𝑝 X ¯ 𝑁 − X(𝑡𝑁 )2𝑝 𝐸 ∣¯ 𝑣 (𝑡, 𝑥) − 𝑣(𝑡, 𝑥)∣2𝑝 ≤ 𝐾𝐸(1 + ∣X(𝑡𝑁 )∣ϰ + ∣X √ √ ¯ 𝑁 − X(𝑡𝑁 )4𝑝 ≤ 𝐶(𝑥)ℎ𝑝 , ¯ 𝑁 ∣ϰ )4𝑝 𝐸 X ≤ 𝐾 𝐸(1 + ∣X(𝑡𝑁 )∣ϰ + ∣X whence (3.4) follows.


2081

Now denote 𝑅 := ∣¯ 𝑣 (𝑡, 𝑥) − 𝑣(𝑡, 𝑥)∣ . The Markov inequality together with (3.4) implies 𝐸𝑅2𝑝 𝑃 (𝑅 > ℎ𝛾 ) ≤ 2𝑝𝛾 ≤ 𝐶(𝑥)ℎ𝑝(1−2𝛾) . ℎ Then for any 𝛾 = 1/2 − 𝜀 there is a sufficiently large 𝑝 ≥ 1 such that (recall that ℎ = 𝑇 /𝑁 ) ( ) ∞ ∞ ∑ ∑ 1 𝑇𝛾 𝑃 𝑅 > 𝛾 ≤ 𝐶(𝑥)𝑇 𝑝(1−2𝛾) < ∞. 𝑝(1−2𝛾) 𝑁 𝑁 𝑁 =1 𝑁 =1 Hence, due to the Borel-Cantelli lemma, the random variable 𝜍 := supℎ>0 ℎ−𝛾 𝑅 is a.s. finite, which implies (3.5). □ 3.2. Weak simulation of the representation (2.6)-(2.7). Now consider another Euler-type scheme for (2.7): (3.7)

[

˜ 0 = 𝑥, 𝑋 ˜ 𝑘+1 = 𝑋 ˜ 𝑘 + ℎ 𝑏(𝑡𝑘 , 𝑋 ˜𝑘 ) − 𝑋 +

𝑞 ∑ 𝑟=1

˜ 𝑘 )Δ𝑘 𝑤𝑟 + ℎ1/2 𝛼𝑟 (𝑡𝑘 , 𝑋

𝑝 ∑

𝑞 ∑

] ˜ 𝑘 )𝛽𝑟 (𝑡𝑘 , 𝑋 ˜𝑘 ) 𝛼𝑟 (𝑡𝑘 , 𝑋

𝑟=1

˜ 𝑘 )𝜉𝑟𝑘 , 𝜎𝑟 (𝑡𝑘 , 𝑋

𝑟=1

˜ 𝑘 )𝑌˜𝑘 + 𝑌˜0 = 1, 𝑌˜𝑘+1 = 𝑌˜𝑘 + ℎ𝑐(𝑡𝑘 , 𝑋

𝑞 ∑

˜ 𝑘 )𝑌˜𝑘 Δ𝑘 𝑤𝑟 , 𝛽𝑟 (𝑡𝑘 , 𝑋

𝑟=1 𝑞 ∑

˜ 𝑘 )𝑌˜𝑘 + 𝑍˜0 = 0, 𝑍˜𝑘+1 = 𝑍˜𝑘 + ℎ𝑓 (𝑡𝑘 , 𝑋

˜ 𝑘 )𝑌˜𝑘 Δ𝑘 𝑤𝑟 , 𝑘 = 0, . . . , 𝑁 − 1, 𝛾𝑟 (𝑡𝑘 , 𝑋

𝑟=1

where Δ𝑘 𝑤𝑟 := 𝑤𝑟 (𝑡𝑘+1 ) − 𝑤𝑟 (𝑡𝑘 ) and 𝜉𝑟𝑘 are i.i.d. random variables with the moments (3.8)

𝐸𝜉 = 0, 𝐸𝜉 2 = 1, 𝐸𝜉 3 = 0, 𝐸𝜉 𝑚 < ∞,

for a sufficiently large integer 𝑚 ≥ 4. For instance, one can take 𝜉 with the law (3.9)

𝑃 (𝜉 = ±1) = 1/2.

We note that here a part of the SDE system (2.7) (which involves the Wiener process 𝑊 (𝑡)) is simulated weakly while the other part (which involves the Wiener process 𝑤(𝑡)) is simulated in the mean-square sense. Let [ ] ˜ 𝑁 )𝑌˜𝑁 + 𝑍˜𝑁 , (3.10) 𝑣˜(𝑡, 𝑥) := 𝐸 𝑤 𝜑(𝑋 ˜ 𝑁 , 𝑌˜𝑁 , 𝑍˜𝑁 are from (3.7). where 𝑋 The same theorem as Theorem 3.1 is valid for 𝑣˜(𝑡, 𝑥) as well, however, with another proof. Theorem 3.2. Let Assumptions 2.1-2.3 hold. The method (3.7), (3.10) satisfies the inequality for 𝑝 ≥ 1 : )1/(2𝑝) ( 2𝑝 ≤ 𝐶(𝑥)ℎ1/2 , (3.11) 𝐸 ∣˜ 𝑣 (𝑡, 𝑥) − 𝑣(𝑡, 𝑥)∣ where 𝐶 does not depend on the discretization step ℎ; i.e., in particular, (3.7), (3.10) is of mean-square order 1/2.

2082


For almost every trajectory 𝑤(⋅) and any 𝜀 > 0 there exists 𝐶(𝑥, 𝜔) > 0 such that ∣˜ 𝑣 (𝑡, 𝑥) − 𝑣(𝑡, 𝑥)∣ ≤ 𝐶(𝑥, 𝜔)ℎ1/2−𝜀 ,

(3.12)

where 𝐶 does not depend on the discretization step ℎ; i.e., the method (3.7), (3.10) converges with order 1/2 − 𝜀 a.s. Proof. We shall prove the inequality )1/(2𝑝) ( ≤ 𝐶(𝑥)ℎ. (3.13) 𝐸 ∣˜ 𝑣 (𝑡, 𝑥) − 𝑣¯(𝑡, 𝑥)∣2𝑝 Due to Theorem 3.1 (see (3.4)), the inequality (3.11) follows from here. The function 𝑣¯(𝑡, 𝑥) (see (3.3)) is introduced for the time layer 𝑡 = 𝑡0 . For any 𝑡𝑖 let us introduce 𝑣¯(𝑡𝑖 , 𝑥). To this aim we use the scheme (3.2) starting at the time ¯ 𝑖 = 𝑥, 𝑌¯𝑖 = 1, 𝑍¯𝑖 = 0. Denote this solution of the scheme by moment 𝑡𝑖 from 𝑋 ¯ 𝑡 ,𝑥 (𝑡𝑘 ), 𝑌¯𝑡 ,𝑥,1 (𝑡𝑘 ), 𝑍¯𝑡 ,𝑥,1,0 (𝑡𝑘 ), 𝑘 = 𝑖, . . . , 𝑁. For instance, 𝑋 ¯ 𝑁 in (3.3) in this 𝑋 𝑖 𝑖 𝑖 ¯𝑁 = 𝑋 ¯ 𝑡 ,𝑥 (𝑡𝑁 ) = 𝑋 ¯ 𝑡,𝑥 (𝑡𝑁 ). We set notation is equal to 𝑋 0 (3.14) ¯ 𝑡 ,𝑥 (𝑡𝑁 )) 𝑌¯𝑡 ,𝑥,1 (𝑡𝑁 ) + 𝑍¯𝑡 ,𝑥,1,0 (𝑡𝑁 )], 𝑖 = 0, . . . , 𝑁, 𝑡𝑁 = 𝑇. 𝑣¯(𝑡𝑖 , 𝑥) = 𝐸 𝑤 [𝜑(𝑋 𝑖 𝑖 𝑖 One can prove that the function 𝑣¯(𝑡𝑖 , 𝑥) is sufficiently smooth in 𝑥 and satisfies (together with its derivatives) the same inequality as the function 𝑣(𝑡, 𝑥) (see the inequality (2.5)). Using the standard technique (see [23, p. 100]), we can write the difference 𝐷𝑁 := 𝑣˜(𝑡, 𝑥) − 𝑣¯(𝑡, 𝑥) in the form (3.15)

[ ] [ ] ˜ 𝑁 )𝑌˜𝑁 + 𝑍˜𝑁 − 𝐸 𝑤 𝜑(𝑋 ¯ 𝑡 ,𝑥 (𝑡𝑁 )) 𝑌¯𝑡 ,𝑥,1 (𝑡𝑁 ) + 𝑍¯𝑡 ,𝑥,1,0 (𝑡𝑁 ) 𝐷𝑁 = 𝐸 𝑤 𝜑(𝑋 0 0 0 =𝐸

𝑤

𝑁 −1 ∑

˜ 𝑖 , 𝑌˜𝑖 , 𝑍˜𝑖 ), 𝜌𝑖 ( 𝑋

𝑖=0

where (3.16)

˜ 𝑖 , 𝑌˜𝑖 , 𝑍˜𝑖 ) = 𝐸 𝑤,𝑋˜ 𝑖 ,𝑌˜𝑖 ,𝑍˜𝑖 [¯ ˜ 𝑖+1 )𝑌˜𝑖+1 + 𝑍˜𝑖+1 𝜌𝑖 ( 𝑋 𝑣 (𝑡𝑖+1 , 𝑋 ¯ ˜ (𝑡𝑖+1 ))𝑌¯ ˜ ˜ (𝑡𝑖+1 ) − 𝑍¯ ˜ ˜ ˜ (𝑡𝑖+1 )], −¯ 𝑣 (𝑡𝑖+1 , 𝑋 𝑡𝑖 ,𝑋𝑖

𝑡𝑖 ,𝑋𝑖 ,𝑌𝑖

𝑡𝑖 ,𝑋𝑖 ,𝑌𝑖 ,𝑍𝑖

˜ 𝑖 ,𝑌˜𝑖 ,𝑍 ˜𝑖 𝑤,𝑋

and 𝐸 [⋅] means the conditional expectation 𝐸[⋅/𝑤(𝑠) − 𝑤(𝑡), 𝑡 ≤ 𝑠 ≤ ˜ 𝑖 , 𝑌˜𝑖 , 𝑍˜𝑖 ]. The presentation (3.15) follows from the equalities 𝑇;𝑋 (3.17) (3.18)

(3.19)

˜ 𝑁 )𝑌˜𝑁 + 𝑍˜𝑁 = 𝜑(𝑋 ˜ 𝑁 )𝑌˜𝑁 + 𝑍˜𝑁 , 𝑣¯(𝑡𝑁 , 𝑋 ¯ ˜ (𝑡1 ))𝑌¯ ˜ ˜ (𝑡1 ) + 𝑍¯ ˜ ˜ ˜ (𝑡1 )] 𝐸 𝑤 [¯ 𝑣 (𝑡1 , 𝑋 𝑡0 ,𝑋0 𝑡0 ,𝑋0 ,𝑌0 𝑡0 ,𝑋0 ,𝑌0 ,𝑍0 [ ] ¯ 𝑡 ,𝑥 (𝑇 ))𝑌¯𝑡 ,𝑥,1 (𝑇 ) + 𝑍¯𝑡 ,𝑥,1,0 (𝑇 ) , = 𝐸 𝑤 𝜑(𝑋 0 0 0 ˜ 𝑘 )𝑌˜𝑘 + 𝑍˜𝑘 ] = 𝐸 𝑤 [¯ ¯ ˜ (𝑡𝑘+1 ))𝑌¯ ˜ 𝑣 (𝑡𝑘 , 𝑋 𝑣 (𝑡𝑘+1 , 𝑋 𝐸 𝑤 [¯ 𝑡𝑘 ,𝑋𝑘

+ 𝑍¯𝑡𝑘 ,𝑋˜ 𝑘 ,𝑌˜𝑘 ,𝑍˜𝑘 (𝑡𝑘+1 )],

𝑡𝑘 ,𝑋𝑘 ,𝑌˜𝑘 (𝑡𝑘+1 )

𝑘 = 1, . . . , 𝑁 − 1. The equality (3.17) is obvious. To prove (3.18) and (3.19), we use the following evident equalities: (3.20) 𝑌¯𝑡 ,𝑥,𝑦 (𝜗) = 𝑌¯𝑡 ,𝑥,1 (𝜗) ⋅ 𝑦, 𝑍¯𝑡 ,𝑥,𝑦,𝑧 (𝜗) = 𝑧 + 𝑍¯𝑡 ,𝑥,1,0 (𝜗) ⋅ 𝑦. 𝑘

𝑘

𝑘

𝑘


2083

We have (see (3.14)) ˜

˜ 𝑘 ) = 𝐸 𝑤,𝑋𝑘 [𝜑(𝑋 ¯ ˜ (𝑇 ))𝑌¯ ˜ (𝑇 ) + 𝑍¯ ˜ 𝑣¯(𝑡𝑘 , 𝑋 𝑡𝑘 ,𝑋𝑘 𝑡𝑘 ,𝑋𝑘 ,1 𝑡𝑘 ,𝑋𝑘 ,1,0 (𝑇 )] ˜ ¯ 𝑡 ,𝑋¯ = 𝐸 𝑤,𝑋𝑘 [𝜑(𝑋 𝑘+1 𝑡

+𝑍¯𝑡𝑘+1 ,𝑋¯ 𝑡

˜ (𝑡𝑘+1 ) 𝑘 ,𝑋𝑘

(𝑇 ))𝑌¯𝑡𝑘+1 ,𝑋¯ 𝑡

¯

¯

˜ (𝑡𝑘+1 ),𝑌𝑡 ,𝑋 ˜ ,1 (𝑡𝑘+1 ) 𝑘 ,𝑋𝑘 𝑘 𝑘

¯

˜ (𝑡𝑘+1 ),𝑌𝑡 ,𝑋 ˜ ,1 (𝑡𝑘+1 ),𝑍𝑡 ,𝑋 ˜ ,1,0 (𝑡𝑘+1 ) 𝑘 ,𝑋𝑘 𝑘 𝑘 𝑘 𝑘

(𝑇 )

(𝑇 )].

Therefore due to (3.20), we get ˜

˜ 𝑘 ) = 𝐸 𝑤,𝑋𝑘 [{𝜑(𝑋 ¯ 𝑡 ,𝑋¯ 𝑣¯(𝑡𝑘 , 𝑋 𝑘+1 𝑡 +𝑍¯𝑡𝑘+1 ,𝑋¯ 𝑡

˜ (𝑡𝑘+1 ),1,0 𝑘 ,𝑋𝑘

˜ (𝑡𝑘+1 ) 𝑘 ,𝑋𝑘

(𝑇 ))𝑌¯𝑡𝑘+1 ,𝑋¯ 𝑡

˜ (𝑡𝑘+1 ),1 𝑘 ,𝑋𝑘

(𝑇 )

(𝑇 )} ⋅ 𝑌¯𝑡𝑘 ,𝑋˜ 𝑘 ,1 (𝑡𝑘+1 ) + 𝑍¯𝑡𝑘 ,𝑋˜ 𝑘 ,1,0 (𝑡𝑘+1 )]

˜

¯ ˜ (𝑡𝑘+1 )) ⋅ 𝑌¯ ˜ (𝑡𝑘+1 ) + 𝑍¯ ˜ = 𝐸 𝑤,𝑋𝑘 [¯ 𝑣 (𝑡𝑘+1, 𝑋 𝑡𝑘 ,𝑋𝑘 𝑡𝑘 ,𝑋𝑘 ,1 𝑡𝑘 ,𝑋𝑘 ,1,0 (𝑡𝑘+1 )]. ˜ 𝑘 ) in 𝐸 𝑤 [¯ ˜ 𝑘 )𝑌˜𝑘 + 𝑍˜𝑘 ] and again using (3.20), we 𝑣 (𝑡𝑘 , 𝑋 Substituting this 𝑣¯(𝑡𝑘 , 𝑋 obtain (3.19). The equality (3.18) is proved analogously. So, the presentation (3.15) is proved. Our next step consists in estimating the smallness of the one-step error. It is easy to see that 𝑍˜𝑖+1 = 𝑍¯𝑡𝑖 ,𝑋˜ 𝑖 ,𝑌˜𝑖 ,𝑍˜𝑖 (𝑡𝑖+1 ) and 𝑌˜𝑖+1 = 𝑌¯𝑡𝑖 ,𝑋˜ 𝑖 ,𝑌˜𝑖 (𝑡𝑖+1 ). Then ˜

˜

˜ 𝑖 , 𝑌˜𝑖 , 𝑍˜𝑖 ) = 𝜌𝑖 (𝑋 ˜ 𝑖 , 𝑌˜𝑖 ) = 𝐸 𝑤,𝑋𝑖 ,𝑌𝑖 𝑌˜𝑖+1 [¯ ˜ 𝑖+1 ) − 𝑣¯(𝑡𝑖+1 , 𝑋 ¯ ˜ (𝑡𝑖+1 ))]. 𝜌𝑖 ( 𝑋 𝑣 (𝑡𝑖+1 , 𝑋 𝑡𝑖 ,𝑋𝑖 ˜ 𝑖+1 ) with respect to powers of Δ ˜ 𝑖 := Now we write the Taylor expansion of 𝑣¯(𝑡𝑖+1 , 𝑋 ˜ ˜ ˜ 𝑋𝑖+1 −𝑋𝑖 in a neighborhood of 𝑋𝑖 and with the Lagrange remainder term containing ¯ ˜ (𝑡𝑖+1 )) with respect to Δ ¯ 𝑖 := terms of order four. We similarly expand 𝑣¯(𝑡𝑖+1 , 𝑋 𝑡𝑖 ,𝑋𝑖 ¯ ˜ (𝑡𝑖+1 ) − 𝑋 ˜ 𝑖 . As a result, we get 𝑋 𝑡𝑖 ,𝑋𝑖 ⎡ ˜ 𝑖 , 𝑌˜𝑖 ) = 𝐸 𝜌𝑖 ( 𝑋

˜ 𝑖 ,𝑌˜𝑖 𝑤,𝑋

1 𝑌˜𝑖+1 ⎣ 24 𝑗

𝑑 ∑

{

∂4𝑣 ˜ ×Δ ˜ 𝑗2 Δ ˜ 𝑗3 Δ ˜ 𝑗4 ˜ 𝑗1 Δ (𝜃) 𝑖 𝑖 𝑖 𝑖 𝑗1 ∂𝑥𝑗2 ∂𝑥𝑗3 ∂𝑥𝑗4 ∂𝑥 1 ,𝑗2 ,𝑗3 ,𝑗4 =1 ⎤ } ∂4𝑣 𝑗 𝑗 𝑗 𝑗 ¯ ×Δ ¯ 2Δ ¯ 3Δ ¯ 4 ⎦, ¯ 1Δ − 𝑗1 𝑗2 𝑗3 𝑗4 (𝜃) 𝑖 𝑖 𝑖 𝑖 ∂𝑥 ∂𝑥 ∂𝑥 ∂𝑥

˜ 𝑖+1 and 𝑋 ˜ 𝑖 , and 𝜃¯ is a point between 𝑋 ¯ ˜ (𝑡𝑖+1 ) and where 𝜃˜ is a point between 𝑋 𝑡𝑖 ,𝑋𝑖 ˜ 𝑖 . Then it is not difficult to obtain that for 𝑝 ≥ 1, 𝑋 (3.21)

˜ 𝑖 , 𝑌˜𝑖 , 𝑍˜𝑖 )∣2𝑝 = 𝑂(ℎ4𝑝 ). 𝐸∣𝜌𝑖 (𝑋

Due to (3.15) and (3.21), we get

2084


(3.22)

2𝑝 𝐸𝐷𝑁

=𝐸

[𝑁 −1 ∑

]2𝑝 ˜ 𝑖 , 𝑌˜𝑖 , 𝑍˜𝑖 ) 𝜌𝑖 ( 𝑋

𝑖=0

≤ 𝑁 2𝑝−1

𝑁 −1 ∑

2𝑝 ˜ 𝑖 , 𝑌˜𝑖 , 𝑍˜𝑖 ) ≤ 𝐶(𝑥)ℎ2𝑝 , 𝐸 𝜌𝑖 (𝑋

𝑖=0

which implies (3.13) (and consequently (3.11)). The inequality (3.12) is proved exactly as its analogue in Theorem 3.1. □ Remark 3.1. In some particular cases of the SPDE (2.1)-(2.2) the order of a.s. convergence of the Euler schemes (3.2), (3.3) and (3.7), (3.10) is higher. For instance, it is not difficult to modify the proofs of Theorems 3.1 and 3.2 to get that the Euler schemes have a.s. order of convergence 1 − 𝜀 if 𝑎 and 𝛼𝑟 are constant and 𝛽𝑟 = 0 and 𝛾𝑟 = 0 (note that in this case (2.7) is a system with additive noise for which the standard Euler scheme for SDEs is of mean-square order 1 [23]). Furthermore, in [25] it is proved that the schemes considered there are also of a.s. order 1 − 𝜀 when 𝛼𝑟 = 0, 𝛾𝑟 = 0, 𝑐 = 0, 𝑓 = 0, 𝛽𝑟 ∕= 0. Remark 3.2. Using the weak-sense numerical integration of SDEs in bounded domains (see [23, Chap. 6] and the references therein), the proposed approach can be carried over to the boundary value problems for SPDEs. 4. Other probabilistic representations, Monte Carlo error, and variance reduction Using the Monte Carlo technique, we approximate the solution of the backward SPDE (2.1)-(2.2) as (see (2.6)): (4.1)

𝑣(𝑡, 𝑥) := 𝑣(𝑡, 𝑥; 𝜔) = 𝐸 𝑤 [𝜑(𝑋𝑡,𝑥 (𝑇 ))𝑌𝑡,𝑥,1 (𝑇 ) + 𝑍𝑡,𝑥,1,0 (𝑇 )] ≈ 𝑣¯(𝑡, 𝑥) := 𝐸 𝑤 [𝜑(𝑋𝑁 )𝑌𝑁 + 𝑍𝑁 ] ≈ 𝑣ˆ(𝑡, 𝑥) :=

𝑀 1 ∑ [𝜑(𝑚 𝑋𝑁 )𝑚 𝑌𝑁 + 𝑀 𝑚=1

𝑚 𝑍𝑁 ] ,

where the first approximate equality involves an error due to replacing 𝑋, 𝑌, 𝑍 by 𝑋𝑁 , 𝑌𝑁 , 𝑍𝑁 (the error of numerical integration of (2.7) by (3.7) or (3.2)) and the error in the second approximate equality comes from the Monte Carlo technique; 𝑚 𝑋𝑁 , 𝑚 𝑌𝑁 , 𝑚 𝑍𝑁 , 𝑚 = 1, . . . , 𝑀, are independent realizations of 𝑋𝑁 , 𝑌𝑁 , 𝑍𝑁 . The error of numerical integration was analyzed in the previous section. Now let us consider the Monte Carlo error. The error of the Monte Carlo method in (4.1) is evaluated by { [ }] ¯ 𝑡,𝑥 (𝑇 ))𝑌¯𝑡,𝑥,1 (𝑇 ) + 𝑍¯𝑡,𝑥,1,0 (𝑇 ) 1/2 𝑉 𝑎𝑟 𝑤 𝜑(𝑋 , 𝜌¯ = 𝑐 𝑀 1/2 where, e.g., the values 𝑐 = 1, 2, 3{correspond to the fiducial probabilities 0.68, 0.95, } ¯ 𝑡,𝑥 (𝑇 ))𝑌¯𝑡,𝑥,1 (𝑇 ) + 𝑍¯𝑡,𝑥,1,0 (𝑇 ) is close to the 0.997, respectively. Since 𝑉 𝑎𝑟 𝑤 𝜑(𝑋 variance 𝑉 := 𝑉 𝑎𝑟 𝑤 [𝜑(𝑋𝑡,𝑥 (𝑇 ))𝑌𝑡,𝑥,1 (𝑇 ) + 𝑍𝑡,𝑥,1,0 (𝑇 )] , we can assume that the error of the Monte Carlo method is estimated by 𝜌=𝑐

𝑉 1/2 . 𝑀 1/2


2085

If 𝑉 is large, then we have to simulate a very large number of trajectories to achieve a satisfactory accuracy. Fortunately, there exist other probabilistic representations for 𝑣(𝑡, 𝑥) which allow us to reduce the variance. The solution of the problem (2.1)-(2.2) also has the following probabilistic representations: (4.2)

𝑣(𝑡, 𝑥) = 𝐸 𝑤 [𝜑(𝑋𝑡,𝑥 (𝑇 ))𝑌𝑡,𝑥,1 (𝑇 ) + 𝑍𝑡,𝑥,1,0 (𝑇 )] , 𝑡0 ≤ 𝑡 ≤ 𝑇,

where 𝑋𝑡,𝑥 (𝑠), 𝑌𝑡,𝑥,𝑦 (𝑠), 𝑍𝑡,𝑥,𝑦,𝑧 (𝑠), 𝑡 ≤ 𝑠 ≤ 𝑇, is the solution of the SDEs [ ] 𝑞 𝑝 ∑ ∑ 𝑑𝑋 = 𝑏(𝑠, 𝑋) − (4.3) 𝛼𝑟 (𝑠, 𝑋)𝛽𝑟 (𝑠, 𝑋) − 𝜎𝑟 (𝑠, 𝑋)𝜇𝑟 (𝑠, 𝑋) 𝑑𝑠 +

𝑝 ∑

𝑟=1

𝜎𝑟 (𝑠, 𝑋)𝑑𝑊𝑟 (𝑠) +

𝑟=1

𝑑𝑌 = 𝑐(𝑠, 𝑋)𝑌 𝑑𝑠 + 𝑑𝑍 = 𝑓 (𝑠, 𝑋)𝑌 𝑑𝑠 +

𝑝 ∑ 𝑟=1 𝑝 ∑

𝑞 ∑

𝑟=1

𝛼𝑟 (𝑠, 𝑋)𝑑𝑤𝑟 (𝑠),

𝑟=1

𝜇𝑟 (𝑠, 𝑋)𝑌 𝑑𝑊𝑟 (𝑠) + 𝜆𝑟 (𝑠, 𝑋)𝑌 𝑑𝑊𝑟 (𝑠) +

𝑟=1

𝑞 ∑ 𝑟=1 𝑞 ∑

𝑋(𝑡) = 𝑥,

𝛽𝑟 (𝑠, 𝑋)𝑌 𝑑𝑤𝑟 (𝑠), 𝑌 (𝑡) = 𝑦, 𝛾𝑟 (𝑠, 𝑋)𝑌 𝑑𝑤𝑟 (𝑠), 𝑍(𝑡) = 𝑧,

𝑟=1

and where 𝜇 = (𝜇1 , . . . , 𝜇𝑝 )⊤ and 𝜆 = (𝜆1 , . . . , 𝜆𝑝 )⊤ are arbitrary 𝑝-dimensional vector functions satisfying some regularity assumptions. When 𝜇 = 0 and 𝜆 = 0, we have the usual representation (2.6)-(2.7). The representation for 𝜇 = 0, 𝜆 ∕= 0 follows from the equality ∫ 𝑇 𝜆⊤ (𝑠, 𝑋(𝑠))𝑌 (𝑠)𝑑𝑊 (𝑠) = 0. 𝐸𝑤 𝑡

For 𝜇 ∕= 0, it can be proved by arguments similar to those in [16, pp. 308-309], making use of Theorem 4.4.5 [16, p. 152] for the forward flow. We should note that 𝑋, 𝑌 , 𝑍 in (4.2)-(4.3) differ from 𝑋, 𝑌 , 𝑍 in (2.6)-(2.7); however, this does not lead to any ambiguity. While 𝑣(𝑡, 𝑥) does not depend on the choice of 𝜇 and 𝜆, the variance 𝑉 = 𝑉 𝑎𝑟 𝑤 [𝜑(𝑋𝑡,𝑥 (𝑇 ))𝑌𝑡,𝑥,1 (𝑇 ) + 𝑍𝑡,𝑥,1,0 (𝑇 )] does. Then one may hope to find 𝜇 and 𝜆 such that the variance 𝑉 is relatively low and thus the Monte Carlo error can be reduced. Theorem 4.1 is helpful in this respect. Its use for variance reduction gives the method of importance sampling if 𝜆 = 0 and the method of control variates if 𝜇 = 0. The form of Theorem 4.1 is very similar to the corresponding well-known result in the case of deterministic PDEs (see [20, 27, 22] and also [23, Section 2.4]). However, its proof is much more complicated in the case of SPDEs than for PDEs as can be seen below. Theorem 4.1. Let Assumptions 2.1-2.3 hold. Let 𝜇𝑟 and 𝜆𝑟 be such that for any 𝑥 ∈ R𝑑 there exists a solution to the system (4.3) on the interval [𝑡, 𝑇 ]. Then

(4.4)

𝑉 𝑎𝑟 𝑤 [𝜑(𝑋𝑡,𝑥 (𝑇 )) 𝑌𝑡,𝑥,1 (𝑇 ) + 𝑍𝑡,𝑥,1,0 (𝑇 )] [ 𝑑 ]2 ∫ 𝑇 𝑝 ∑ ∑ ∂𝑣 2 = 𝐸𝑤 𝑌𝑡,𝑥,1 𝜎𝑟𝑖 𝑖 + 𝜇𝑟 𝑣 + 𝜆𝑟 𝑑𝜃 ∂𝑥 𝑡 𝑟=1 𝑖=1

provided that the expectation in (4.4) exists. In (4.4) all the functions in the integrand have (𝜃, 𝑋𝑡,𝑥 (𝜃)) as their argument.

2086


Proof. It is convenient to introduce the notation 𝐸𝑠𝑤 (⋅) := 𝐸 (⋅/𝑤(𝑠′ ) − 𝑤(𝑠), 𝑠 ≤ 𝑠′ ≤ 𝑇 ) , (4.5)

𝑣(𝑠, 𝑥) = 𝑣(𝑠, 𝑥; 𝜔) := 𝐸𝑠𝑤 [𝜑(𝑋𝑠,𝑥 (𝑇 ))𝑌𝑠,𝑥,1 (𝑇 ) + 𝑍𝑠,𝑥,1,0 (𝑇 )] := 𝐸 [𝜑(𝑋𝑠,𝑥 (𝑇 ))𝑌𝑠,𝑥,1 (𝑇 ) + 𝑍𝑠,𝑥,1,0 (𝑇 )/𝑤(𝑠′ ) − 𝑤(𝑠), 𝑠 ≤ 𝑠′ ≤ 𝑇 ] .

Clearly, 𝐸𝑡𝑤 (⋅) = 𝐸 𝑤 (⋅) . The notation 𝑉 𝑎𝑟𝑠𝑤 (⋅) can be introduced analogously. We get from (4.5): (4.6)

𝑣(𝑠, 𝑋𝑡,𝑥 (𝑠)) = 𝐸[𝜑(𝑋𝑠,𝑋𝑡,𝑥 (𝑠) (𝑇 ))𝑌𝑠,𝑋𝑡,𝑥 (𝑠),1 (𝑇 ) + 𝑍𝑠,𝑋𝑡,𝑥 (𝑠),1,0 (𝑇 )/ 𝑊 (𝑠′ ) − 𝑊 (𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑠; 𝑤(𝑠′ ) − 𝑤(𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑇 ].

The presentation (4.6) follows from the assertion: let ℱ˜ and ℱ ′ be independent 𝜎-algebras and ℱ˜ ∨ ℱ ′ be the minimal 𝜎-algebra generated by ℱ˜ and ℱ ′ ; if 𝜉 is ℱ˜ -measurable, 𝑓 (𝑥; 𝜔) does not depend on ℱ˜ , and 𝐸(𝑓 (𝑥; 𝜔)/ℱ ′ ) = 𝜓(𝑥; 𝜔), then 𝐸(𝑓 (𝜉; 𝜔)/ℱ˜ ∨ ℱ ′ ) = 𝜓(𝜉; 𝜔) (for the trivial ℱ ′ this assertion can be found, e.g., in [6, §10, Lemma 1], [4, Proposition 1.12], which proof can be straightforwardly generalized for a general ℱ ′ ). In the case of (4.6) we have 𝜉 = 𝑋𝑡,𝑥 (𝑠), ℱ˜ is the minimal 𝜎-algebra generated by {𝑊 (𝑠′ )−𝑊 (𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑠; 𝑤(𝑠′ )−𝑤(𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑠}, ℱ ′ is the minimal 𝜎-algebra generated by {𝑤(𝑠′ ) − 𝑤(𝑠), 𝑠 ≤ 𝑠′ ≤ 𝑇 }, and 𝑓 (𝑥; 𝜔) = 𝑣(𝑠, 𝑥; 𝜔). Furthermore, we obtain 𝐸 [𝑣(𝑠, 𝑋𝑡,𝑥 (𝑠))𝑌𝑡,𝑥,1 (𝑠) + 𝑍𝑡,𝑥,1,0 (𝑠)/𝑤(𝑠′ ) − 𝑤(𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑇 ] = 𝐸[𝐸(𝜑(𝑋𝑠,𝑋𝑡,𝑥 (𝑠) (𝑇 ))𝑌𝑠,𝑋𝑡,𝑥 (𝑠),1 (𝑇 ) + 𝑍𝑠,𝑋𝑡,𝑥 (𝑠),1,0 (𝑇 )/ 𝑊 (𝑠′ ) − 𝑊 (𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑠; 𝑤(𝑠′ ) − 𝑤(𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑇 ) × 𝑌𝑡,𝑥,1 (𝑠) + 𝑍𝑡,𝑥,1,0 (𝑠)/𝑤(𝑠′ ) − 𝑤(𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑇 ] = 𝐸[𝐸(𝜑(𝑋𝑡,𝑥 (𝑇 ))𝑌𝑡,𝑥,1 (𝑇 ) + 𝑍𝑡,𝑥,1,0 (𝑇 )/𝑊 (𝑠′ ) − 𝑊 (𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑠; 𝑤(𝑠′ ) − 𝑤(𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑇 )/𝑤(𝑠′ ) − 𝑤(𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑇 ] = 𝐸[𝜑(𝑋𝑡,𝑥 (𝑇 ))𝑌𝑡,𝑥,1 (𝑇 ) + 𝑍𝑡,𝑥,1,0 (𝑇 )/𝑤(𝑠′ ) − 𝑤(𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑇 ] = 𝑣(𝑡, 𝑥). Thus, we prove that for any 𝑠, 𝑡 ≤ 𝑠 ≤ 𝑇, the following formula holds: (4.7)

𝑣(𝑡, 𝑥) = 𝐸𝑡𝑤 [𝑣(𝑠, 𝑋𝑡,𝑥 (𝑠))𝑌𝑡,𝑥,1 (𝑠) + 𝑍𝑡,𝑥,1,0 (𝑠)] , 𝑡 ≤ 𝑠.

Introduce the auxiliary function: 𝑈 (𝑡, 𝑥, 𝑦, 𝑧) := 𝑣(𝑡, 𝑥)𝑦 + 𝑧. Partition the interval [𝑡, 𝑇 ] with the time step ℎ and present the variance 𝑉 in the following way (see a similar recipe in [29]): 𝑉 = 𝑉 𝑎𝑟𝑡𝑤 [𝜑(𝑋𝑡,𝑥 (𝑇 ))𝑌𝑡,𝑥,1 (𝑇 ) + 𝑍𝑡,𝑥,1,0 (𝑇 )] 2

= 𝐸𝑡𝑤 [𝑈 (𝑇, 𝑋𝑡,𝑥 (𝑇 ), 𝑌𝑡,𝑥,1 (𝑇 ), 𝑍𝑡,𝑥,1,0 (𝑇 )) − 𝑈 (𝑡, 𝑥, 1, 0)] [𝑁 −1 ∑ 𝑤 = 𝐸𝑡 {𝑈 (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘+1 ), 𝑌𝑡,𝑥,1 (𝑡𝑘+1 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘+1 )) 𝑘=0

]2 − 𝑈 (𝑡𝑘 , 𝑋𝑡,𝑥 (𝑡𝑘 ), 𝑌𝑡,𝑥,1 (𝑡𝑘 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘 ))}

.


Using (4.7), we get 𝐸(𝑈 (𝑡𝑛+1 , 𝑋𝑡,𝑥 (𝑡𝑛+1 ), 𝑌𝑡,𝑥,1 (𝑡𝑛+1 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑛+1 ))/𝑤(𝑠′ ) − 𝑤(𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑇 ; 𝑊 (𝑠′′ ) − 𝑊 (𝑡), 𝑡 ≤ 𝑠′′ ≤ 𝑡𝑛 ) = 𝑌𝑡,𝑥,1 (𝑡𝑛 ) ( ×𝐸 𝑈 (𝑡𝑛+1 , 𝑋𝑡𝑛 ,𝑋𝑡,𝑥 (𝑡𝑛 ) (𝑡𝑛+1 ), 𝑌𝑡𝑛 ,𝑋𝑡,𝑥 (𝑡𝑛 ),1 (𝑡𝑛+1 ), 𝑍𝑡𝑛 ,𝑋𝑡,𝑥 (𝑡𝑛 ),1,0 (𝑡𝑛+1 )) /𝑤(𝑠′ ) − 𝑤(𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑇 ; 𝑊 (𝑠′′ ) − 𝑊 (𝑡), 𝑡 ≤ 𝑠′′ ≤ 𝑡𝑛 ) + 𝑍𝑡,𝑥,1,0 (𝑡𝑛 ) = 𝑌𝑡,𝑥,1 (𝑡𝑛 )𝑣(𝑡𝑛 , 𝑋𝑡,𝑥 (𝑡𝑛 )) + 𝑍𝑡,𝑥,1,0 (𝑡𝑛 ) = 𝑈 (𝑡𝑛 , 𝑋𝑡,𝑥 (𝑡𝑛 ), 𝑌𝑡,𝑥,1 (𝑡𝑛 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑛 )), whence for 𝑘 < 𝑛, 𝐸𝑡𝑤 ([𝑈 (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘+1 ), 𝑌𝑡,𝑥,1 (𝑡𝑘+1 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘+1 )) − 𝑈 (𝑡𝑘 , 𝑋𝑡,𝑥 (𝑡𝑘 ), 𝑌𝑡,𝑥,1 (𝑡𝑘 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘 ))] × [𝑈 (𝑡𝑛+1 , 𝑋𝑡,𝑥 (𝑡𝑛+1 ), 𝑌𝑡,𝑥,1 (𝑡𝑛+1 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑛+1 )) − 𝑈 (𝑡𝑛 , 𝑋𝑡,𝑥 (𝑡𝑛 ), 𝑌𝑡,𝑥,1 (𝑡𝑛 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑛 )))]) 𝑤 𝐸𝑡 ([𝑈 (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘+1 ), 𝑌𝑡,𝑥,1 (𝑡𝑘+1 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘+1 ))

=

− 𝑈 (𝑡𝑘 , 𝑋𝑡,𝑥 (𝑡𝑘 ), 𝑌𝑡,𝑥,1 (𝑡𝑘 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘 ))] 𝐸(𝑈 (𝑡𝑛+1 , 𝑋𝑡,𝑥 (𝑡𝑛+1 ), 𝑌𝑡,𝑥,1 (𝑡𝑛+1 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑛+1 )) − 𝑈 (𝑡𝑛 , 𝑋𝑡,𝑥 (𝑡𝑛 ), 𝑌𝑡,𝑥,1 (𝑡𝑛 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑛 )) ′

/𝑤(𝑠 ) − 𝑤(𝑡), 𝑡 ≤ 𝑠′ ≤ 𝑇 ; 𝑊 (𝑠′′ ) − 𝑊 (𝑡), 𝑡 ≤ 𝑠′′ ≤ 𝑡𝑛 )) = 0. Then (4.8)

𝑉 =

𝑁 −1 ∑

𝐸𝑡𝑤 {𝑈 (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘+1 ), 𝑌𝑡,𝑥,1 (𝑡𝑘+1 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘+1 ))

𝑘=0

− 𝑈 (𝑡𝑘 , 𝑋𝑡,𝑥 (𝑡𝑘 ), 𝑌𝑡,𝑥,1 (𝑡𝑘 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘 ))}2 . We rewrite the terms under the expectation in (4.8) as (4.9)

𝑈 (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘+1 ), 𝑌𝑡,𝑥,1 (𝑡𝑘+1 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘+1 )) − 𝑈 (𝑡𝑘 , 𝑋𝑡,𝑥 (𝑡𝑘 ), 𝑌𝑡,𝑥,1 (𝑡𝑘 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘 )) = [𝑈 (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘+1 ), 𝑌𝑡,𝑥,1 (𝑡𝑘+1 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘+1 )) − 𝑈 (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘 ), 𝑌𝑡,𝑥,1 (𝑡𝑘 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘 ))] + [𝑈 (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘 ), 𝑌𝑡,𝑥,1 (𝑡𝑘 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘 )) − 𝑈 (𝑡𝑘 , 𝑋𝑡,𝑥 (𝑡𝑘 ), 𝑌𝑡,𝑥,1 (𝑡𝑘 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘 ))] = [𝑈 (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘+1 ), 𝑌𝑡,𝑥,1 (𝑡𝑘+1 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘+1 )) − 𝑈 (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘 ), 𝑌𝑡,𝑥,1 (𝑡𝑘 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘 ))] + 𝑌𝑡,𝑥,1 (𝑡𝑘 ) [𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘 )) − 𝑣(𝑡𝑘 , 𝑋𝑡,𝑥 (𝑡𝑘 ))] .

Let Λ𝑟 𝑣(𝑡, 𝑥) :=

𝑑 ∑ 𝑖=1

𝜎𝑟𝑖 (𝑡, 𝑥)

∂ 𝑣(𝑡, 𝑥). ∂𝑥𝑖

2087

2088


Applying the Ito formula to the first term on the right-hand side of (4.9), we get (4.10)

𝑈 (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘+1 ), 𝑌𝑡,𝑥,1 (𝑡𝑘+1 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘+1 )) ∫ = + +

− 𝑈 (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘 ), 𝑌𝑡,𝑥,1 (𝑡𝑘 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘 )) 𝑡𝑘+1

𝑌𝑡,𝑥,1 (𝜃) [ℒ𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) 𝑡𝑘 𝑞 ∫ 𝑡𝑘+1 ∑ 𝑟=1 𝑡𝑘 𝑝 ∫ 𝑡𝑘+1 ∑ 𝑡𝑘

𝑟=1

+ 𝑓 (𝜃, 𝑋𝑡,𝑥 (𝜃))] 𝑑𝜃

𝑌𝑡,𝑥,1 (𝜃) [ℳ𝑟 𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝛾𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃)))] 𝑑𝑤𝑟 (𝜃) 𝑌𝑡,𝑥,1 (𝜃)[Λ𝑟 𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝜇𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃))

+ 𝜆𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))]𝑑𝑊𝑟 (𝜃). The coefficients of the operators ℒ and ℳ𝑟 in (4.10) have (𝜃, 𝑋𝑡,𝑥 (𝜃)) as their argument. We note (see [29]) that the use of the Ito formula here is legitimate since 𝑡 𝑣(𝑡𝑘+1 , 𝑥) is ℱ𝑇𝑘+1 -adapted and independent of ℱ𝜃 , 𝜃 ≤ 𝑡𝑘+1 . Further, since 𝑣(𝑡, 𝑥) satisfies the SPDE (2.1), we can present the second term on the right-hand side of (4.9) as (4.11)

𝑌𝑡,𝑥,1 (𝑡𝑘 ) [𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘 )) − 𝑣(𝑡𝑘 , 𝑋𝑡,𝑥 (𝑡𝑘 ))] [∫ 𝑡𝑘+1 = −𝑌𝑡,𝑥,1 (𝑡𝑘 ) [(ℒ𝑣)(𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 )) + 𝑓 (𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 ))] 𝑑𝜃 +

𝑞 ∫ 𝑡𝑘+1 ∑ 𝑡𝑘

𝑟=1

𝑡𝑘

]

[(ℳ𝑟 𝑣)(𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 )) + 𝛾𝑟 (𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 )))] ∗ 𝑑𝑤𝑟 (𝜃) .

We note that all the functions in the integrands in (4.11) have (𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 )) as their argument. Using (4.9)-(4.11), properties of Ito integrals, and the independence of 𝑤 and 𝑊, we obtain (4.12) 𝐸𝑡𝑤 {𝑈 (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘+1 ), 𝑌𝑡,𝑥,1 (𝑡𝑘+1 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘+1 )) − 𝑈 (𝑡𝑘 , 𝑋𝑡,𝑥 (𝑡𝑘 ), 𝑌𝑡,𝑥,1 (𝑡𝑘 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘 ))}2 [∫ 𝑡𝑘+1 𝑤 = 𝐸𝑡 {𝑌𝑡,𝑥,1 (𝜃) [ℒ𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝑓 (𝜃, 𝑋𝑡,𝑥 (𝜃))] 𝑡𝑘

− 𝑌𝑡,𝑥,1 (𝑡𝑘 ) [(ℒ𝑣)(𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 )) + 𝑓 (𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 ))]}𝑑𝜃 𝑞 {∫ 𝑡𝑘+1 ∑ 𝑌𝑡,𝑥,1 (𝜃) [ℳ𝑟 𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝛾𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃)))] 𝑑𝑤𝑟 (𝜃) + 𝑟=1

𝑡𝑘

−𝑌𝑡,𝑥,1 (𝑡𝑘 ) + 2𝐸𝑡𝑤

𝑝 ∫ 𝑡𝑘+1 ∑ 𝑟=1

𝑡𝑘

∫

𝑡𝑘+1

𝑡𝑘

}]2 [(ℳ𝑟 𝑣)(𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 )) + 𝛾𝑟 (𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 )))] ∗ 𝑑𝑤𝑟 (𝜃)

𝑌𝑡,𝑥,1 (𝜃)

× [Λ𝑟 𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝜇𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝜆𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))] 𝑑𝑊𝑟 (𝜃)


[∫ ×

𝑡𝑘+1

𝑡𝑘

𝑌𝑡,𝑥,1 (𝜃) [ℒ𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝑓 (𝜃, 𝑋𝑡,𝑥 (𝜃))] 𝑑𝜃

𝑞 ∫ ∑

+

𝑟=1 𝑝 ∑

+

2089

𝑡𝑘+1 𝑡𝑘

∫

𝐸𝑡𝑤

] 𝑌𝑡,𝑥,1 (𝜃) [ℳ𝑟 𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝛾𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃)))] 𝑑𝑤𝑟 (𝜃) 𝑡𝑘+1

𝑡𝑘

𝑟=1

2 𝑌𝑡,𝑥,1 (𝜃)[Λ𝑟 𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝜇𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃))

+ 𝜆𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))]2 𝑑𝜃. In this proof let us use the notation 𝑂(ℎ𝑞 ) for random variables which tend to zero at least as ℎ𝑞 a.s. when ℎ → 0. We have for any 0 < 𝜀 < 1/2, 𝑤𝑟 (𝑠 + ℎ) − 𝑤𝑟 (𝑠) = 𝑂(ℎ1/2−𝜀 ), 𝑊𝑟 (𝑠 + ℎ) − 𝑊𝑟 (𝑠) = 𝑂(ℎ1/2−𝜀 ) 𝑎.𝑠., and also

𝑣(𝑠 + ℎ, 𝑥) − 𝑣(𝑠, 𝑥) = 𝑂(ℎ1/2−𝜀 ) 𝑎.𝑠. One can show that for 0 < 𝜀 < 1/2, ∫ 𝑡𝑘+1 𝑌𝑡,𝑥,1 (𝜃) [ℒ𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝑓 (𝜃, 𝑋𝑡,𝑥 (𝜃))] 𝑡𝑘

− 𝑌𝑡,𝑥,1 (𝑡𝑘 ) [(ℒ𝑣)(𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 )) + 𝑓 (𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 ))]}𝑑𝜃 = 𝑂(ℎ), ∫ 𝑡𝑘+1 𝑌𝑡,𝑥,1 (𝜃) [ℳ𝑟 𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝛾𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃)))] 𝑑𝑤𝑟 (𝜃) 𝑡𝑘

∫

= 𝑌𝑡,𝑥,1 (𝑡𝑘 ) = 𝑌𝑡,𝑥,1 (𝑡𝑘 ) = 𝑌𝑡,𝑥,1 (𝑡𝑘 ) 𝐸𝑡𝑤

∫

𝑡𝑘

∫ ×

𝑡𝑘+1

𝑡𝑘+1

𝑡𝑘 ∫ 𝑡𝑘+1 𝑡𝑘 ∫ 𝑡𝑘+1 𝑡𝑘

[(ℳ𝑟 𝑣)(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘 )) + 𝛾𝑟 (𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 )))] 𝑑𝑤𝑟 (𝜃) + 𝑂(ℎ1−𝜀 ) [(ℳ𝑟 𝑣)(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘 )) + 𝛾𝑟 (𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 )))] ∗ 𝑑𝑤𝑟 (𝜃) + 𝑂(ℎ1−𝜀 ) [(ℳ𝑟 𝑣)(𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 )) + 𝛾𝑟 (𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 )))] ∗ 𝑑𝑤𝑟 (𝜃) + 𝑂(ℎ1−𝜀 ),

𝑌𝑡,𝑥,1 (𝜃)[Λ𝑟 𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝜇𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝜆𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))]𝑑𝑊𝑟 (𝜃)

𝑡𝑘+1

𝑡𝑘

𝑌𝑡,𝑥,1 (𝜃) [ℒ𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝑓 (𝜃, 𝑋𝑡,𝑥 (𝜃))] 𝑑𝜃

= 𝑂(ℎ3/2−𝜀 ), ∫ 𝑡𝑘+1 𝐸𝑡𝑤 𝑌𝑡,𝑥,1 (𝜃)[Λ𝑟 𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝜇𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) 𝑡𝑘

+ 𝜆𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))]𝑑𝑊𝑟 (𝜃) ∫ 𝑡𝑘+1 × 𝑌𝑡,𝑥,1 (𝜃) [ℳ𝑟 𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝛾𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃)))] 𝑑𝑤𝑟 (𝜃) 𝑡𝑘

=

𝐸𝑡𝑤

∫

𝑡𝑘+1

𝑡𝑘

𝑌𝑡,𝑥,1 (𝜃)[Λ𝑟 𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝜇𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃))

+ 𝜆𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))]𝑑𝑊𝑟 (𝜃)

2090


∫ ×

𝑡𝑘+1

𝑡𝑘

𝑌𝑡,𝑥,1 (𝑡𝑘 ) [(ℳ𝑟 𝑣) (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘 )) + 𝛾𝑟 (𝜃, 𝑋𝑡,𝑥 (𝑡𝑘 )))] 𝑑𝑤𝑟 (𝜃)

+𝑂(ℎ3/2−𝜀 ) = 𝑂(ℎ3/2−𝜀 ) , ∫ 𝑡𝑘+1 2 𝐸𝑡𝑤 𝑌𝑡,𝑥,1 (𝜃)[Λ𝑟 𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) + 𝜇𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))𝑣(𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝜃)) 𝑡𝑘

= 𝐸𝑡𝑤

∫

+ 𝜆𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))]2 𝑑𝜃 𝑡𝑘+1

𝑡𝑘

2 𝑌𝑡,𝑥,1 (𝜃)[(Λ𝑟 𝑣)(𝜃, 𝑋𝑡,𝑥 (𝜃)) + 𝜇𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))𝑣(𝜃, 𝑋𝑡,𝑥 (𝜃))

+ 𝜆𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))]2 𝑑𝜃 + 𝑂(ℎ3/2−𝜀 ). Substituting the above relations in the right-hand side of (4.12), we obtain 𝐸𝑡𝑤 {𝑈 (𝑡𝑘+1 , 𝑋𝑡,𝑥 (𝑡𝑘+1 ), 𝑌𝑡,𝑥,1 (𝑡𝑘+1 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘+1 ))

=

𝑝 ∑

𝐸𝑡𝑤

𝑟=1

∫

− 𝑈 (𝑡𝑘 , 𝑋𝑡,𝑥 (𝑡𝑘 ), 𝑌𝑡,𝑥,1 (𝑡𝑘 ), 𝑍𝑡,𝑥,1,0 (𝑡𝑘 ))}2 𝑡𝑘+1

𝑡𝑘

2 𝑌𝑡,𝑥,1 (𝜃)[(Λ𝑟 𝑣)(𝜃, 𝑋𝑡,𝑥 (𝜃)) + 𝜇𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))𝑣(𝜃, 𝑋𝑡,𝑥 (𝜃))

+ 𝜆𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))]2 𝑑𝜃 + 𝑂(ℎ3/2−𝜀 ), which substitution in (4.8) gives ∫ 𝑇 𝑝 ∑ 𝑤 2 𝐸𝑡 𝑌𝑡,𝑥,1 (𝜃)[(Λ𝑟 𝑣)(𝜃, 𝑋𝑡,𝑥 (𝜃)) + 𝜇𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))𝑣(𝜃, 𝑋𝑡,𝑥 (𝜃)) 𝑉 = 𝑟=1

𝑡

+ 𝜆𝑟 (𝜃, 𝑋𝑡,𝑥 (𝜃))]2 𝑑𝜃 +

𝑁 −1 ∑

𝑂(ℎ3/2−𝜀 ),

𝑘=0

and tending ℎ to 0 we arrive at (4.4).

□

Theorem 4.1 implies that if 𝜇𝑟 and 𝜆𝑟 are such that (4.13)

𝑑 ∑ 𝑖=1

𝜎𝑟𝑖

∂𝑣 + 𝜇𝑟 𝑣 + 𝜆𝑟 = 0, 𝑟 = 1, . . . , 𝑝, ∂𝑥𝑖

then the right-hand side of (4.4) is zero and, consequently, the variance is zero. We recall that 𝑣(𝑠, 𝑥) depends on 𝑤(𝜃) − 𝑤(𝑠), 𝑠 ≤ 𝜃 ≤ 𝑇. However, we need to require that 𝜇𝑟 (𝑠, 𝑥) does not depend on 𝑤(𝜃) − 𝑤(𝑠), 𝑠 ≤ 𝜃 ≤ 𝑇 ; otherwise, the coefficients in (4.3) depend on 𝑤(𝜃) − 𝑤(𝑠), 𝑠 ≤ 𝜃 ≤ 𝑇, and we are facing the difficulty with interpreting (4.3) except the case when 𝛼𝑟 is independent of 𝑥 and 𝛽𝑟 = 𝛾𝑟 = 0. In the case mentioned, all the integrals appearing in the integral form of (4.3) can be given the usual meaning in the sense of Ito calculus. At the same time, dependence of the function 𝜆𝑟 (𝑠, 𝑥) on 𝑤(𝜃) − 𝑤(𝑠), 𝑠 ≤ 𝜃 ≤ 𝑇, does not cause any trouble in interpreting the third equation in (4.3), and the identity (4.13) can, in principle, be reached. We note that Theorem 4.1 and its proof are valid for 𝜆𝑟 (𝑠, 𝑥) depending on 𝑤(𝜃) − 𝑤(𝑠), 𝑠 ≤ 𝜃 ≤ 𝑇. Furthermore, when (4.3) is discretely simulated, one can simulate the process 𝑤(𝑠) in advance and fix it. Then it is possible to get approximate solutions of (4.3) for 𝜇𝑟 depending on a current position and on the fixed 𝑤(𝜃) − 𝑤(𝑠), 𝑠 ≤ 𝜃 ≤ 𝑇 . Thus we obtain a heuristic interpretation of (4.3) for


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𝜇𝑟 depending on the future of 𝑤 and, apparently, the result of Theorem 4.1 can be used for such 𝜇𝑟 as well. Of course, 𝜇𝑟 and 𝜆𝑟 satisfying (4.13) cannot be constructed without knowing 𝑣(𝑠, 𝑥), 𝑡 ≤ 𝑠 ≤ 𝑇, 𝑥 ∈ R𝑑 . Nevertheless, Theorem 4.1 claims a general possibility of variance reduction by a proper choice of the functions 𝜇𝑟 and 𝜆𝑟 . Theorem 4.1 can be used, for example, if we know a function 𝑣˜(𝑠, 𝑥) being close to 𝑣(𝑠, 𝑥) (see a practical approach to constructing such approximate functions in the case of deterministic PDEs in [26]). Then we take any 𝜇𝑟 and 𝜆𝑟 satisfying (4.14)

𝑑 ∑ 𝑖=1

𝜎𝑟𝑖

∂˜ 𝑣 + 𝜇𝑟 𝑣˜ + 𝜆𝑟 = 0, ∂𝑥𝑖

and we expect that the variance 𝑉 𝑎𝑟 𝑤 [𝜑(𝑋𝑡,𝑥 (𝑇 ))𝑌𝑡,𝑥,1 (𝑇 ) + 𝑍𝑡,𝑥,1,0 (𝑇 )] is small although not zero.

5. Layer methods for linear SPDEs In the previous sections we use the averaging over the characteristic formula to propose Monte Carlo methods for linear SPDEs. In this section we exploit probabilistic representations to construct some layer methods for linear SPDEs (see them in the case of deterministic PDEs in [21, 23]). Layer methods for semilinear SPDEs are considered in the next section. The layer methods are competitive with finite difference schemes [10, 34]. They can be used for relatively low-dimensional SPDEs when one needs to find the solution 𝑣(𝑡, 𝑥) everywhere in [𝑡0 , 𝑇 ] × R𝑑 . The bottleneck for using the layer methods in the case of higher-dimensional SPDEs is interpolation, which should be used to restore the function 𝑣¯(𝑡𝑘 , 𝑥), 𝑥 ∈ R𝑑 , by its values 𝑣¯(𝑡𝑘 , 𝑥𝑗 ) at a set of nodes 𝑥𝑗 . The linear interpolation, which is successfully applied for low-dimensional problems, is not effective for the higherdimensional ones. However, layer methods can be turned into numerical algorithms by exploiting other approximations of functions. New developments in the theory of multidimensional approximation (see, e.g., [33] and the references therein) give approximations which can work well in relatively high dimensions. The method (3.7), (3.10) can be turned into a layer method. Indeed, analogously to the probabilistic representation (2.6)-(2.7) one can write down the local probabilistic representation of the solution to (2.1)-(2.2) (see (4.7)): (5.1)

𝑣(𝑡𝑘 , 𝑥) = 𝐸 𝑤 [𝑣(𝑡𝑘+1 , 𝑋𝑡𝑘 ,𝑥 (𝑡𝑘+1 ))𝑌𝑡𝑘 ,𝑥,1 (𝑡𝑘+1 ) + 𝑍𝑡𝑘 ,𝑥,1,0 (𝑡𝑘+1 )] ,

where 𝑋𝑡,𝑥 (𝑠), 𝑌𝑡,𝑥,𝑦 (𝑠), 𝑍𝑡,𝑥,𝑦,𝑧 (𝑠), 𝑡 ≤ 𝑠 ≤ 𝑇, is the solution of the SDEs (2.7). Replacing 𝑋𝑡𝑘 ,𝑥 (𝑡𝑘+1 ), 𝑌𝑡𝑘 ,𝑥,1 (𝑡𝑘+1 ), 𝑍𝑡𝑘 ,𝑥,1,0 (𝑡𝑘+1 ), by the one-step Euler approx¯ 𝑡 ,𝑥 (𝑡𝑘+1 ), 𝑌¯𝑡 ,𝑥,1 (𝑡𝑘+1 ), 𝑍¯𝑡 ,𝑥,1,0 (𝑡𝑘+1 ) as in (3.7), (3.9), we obtain the imation 𝑋 𝑘 𝑘 𝑘 one-step approximation 𝑣˘(𝑡𝑘 , 𝑥) for 𝑣(𝑡𝑘 , 𝑥) : [ ] ¯ 𝑡 ,𝑥 (𝑡𝑘+1 ))𝑌¯𝑡 ,𝑥,1 (𝑡𝑘+1 ) + 𝑍¯𝑡 ,𝑥,1,0 (𝑡𝑘+1 ) . (5.2) 𝑣˘(𝑡𝑘 , 𝑥) = 𝐸 𝑤 𝑣(𝑡𝑘+1 , 𝑋 𝑘 𝑘 𝑘 Now for simplicity in writing let us consider the case 𝑑 = 1 and 𝑞 = 1. Thanks to the use of the discrete random variables (3.9), the expectation in (5.2) can be evaluated exactly:

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(5.3)


{

) 1 ( 𝑣 𝑡𝑘+1 , 𝑥 + ℎ [𝑏(𝑡𝑘 , 𝑥)−𝛼(𝑡𝑘 , 𝑥)𝛽(𝑡𝑘 , 𝑥)]+ℎ1/2 𝜎(𝑡𝑘 , 𝑥)+𝛼(𝑡𝑘 , 𝑥)Δ𝑘 𝑤 2 ( )} 1 + 𝑣 𝑡𝑘+1 , 𝑥 + ℎ [𝑏(𝑡𝑘 , 𝑥) − 𝛼(𝑡𝑘 , 𝑥)𝛽(𝑡𝑘 , 𝑥)] − ℎ1/2 𝜎(𝑡𝑘 , 𝑥) + 𝛼(𝑡𝑘 , 𝑥)Δ𝑘 𝑤 2 × [1 + ℎ𝑐(𝑡𝑘 , 𝑥) + 𝛽(𝑡𝑘 , 𝑥)Δ𝑘 𝑤] + ℎ𝑓 (𝑡𝑘 , 𝑥) + 𝛾(𝑡𝑘 , 𝑥)Δ𝑘 𝑤.

𝑣˘(𝑡𝑘 , 𝑥) =

Based on the one-step approximation (5.3), we obtain the layer method for (2.1)(2.2) with 𝑑 = 1, 𝑞 = 1: 𝑣˜(𝑡𝑁 , 𝑥) = 𝜑(𝑥), ) 1 ( 𝑣˜(𝑡𝑘 , 𝑥) = 𝑣˜ 𝑡𝑘+1 , 𝑥 + ℎ [𝑏(𝑡𝑘 , 𝑥)−𝛼(𝑡𝑘 , 𝑥)𝛽(𝑡𝑘 , 𝑥)]+ℎ1/2 𝜎(𝑡𝑘 , 𝑥)+𝛼(𝑡𝑘 , 𝑥)Δ𝑘 𝑤 2 ) 1 ( + 𝑣˜ 𝑡𝑘+1 , 𝑥 + ℎ [𝑏(𝑡𝑘 , 𝑥) − 𝛼(𝑡𝑘 , 𝑥)𝛽(𝑡𝑘 , 𝑥)] − ℎ1/2 𝜎(𝑡𝑘 , 𝑥) + 𝛼(𝑡𝑘 , 𝑥)Δ𝑘 𝑤 2 × [1 + ℎ𝑐(𝑡𝑘 , 𝑥) + 𝛽(𝑡𝑘 , 𝑥)Δ𝑘 𝑤] + ℎ𝑓 (𝑡𝑘 , 𝑥) + 𝛾(𝑡𝑘 , 𝑥)Δ𝑘 𝑤, 𝑘 = 𝑁 − 1, . . . , 0.

(5.4)

{

It is not difficult to see that 𝑣˜(𝑡𝑘 , 𝑥) from (5.4) coincides with the 𝑣˜(𝑡𝑘 , 𝑥) due to (3.10). Then according to Theorem 3.2, the estimates (3.11) and (3.12) are valid for 𝑣˜(𝑡𝑘 , 𝑥) from (5.4). To realize (5.4) numerically, it is sufficient to calculate the functions 𝑣˜(𝑡𝑘 , 𝑥) at some knots 𝑥𝑗 applying some kind of interpolation at every layer. So, to become a numerical algorithm, the layer method (5.4) needs a discretization in the variable 𝑥. Consider the equidistant space discretization: (5.5)

𝑥𝑗 = 𝑥0 + 𝑗ℎ𝑥 , 𝑗 = 0, ±1, ±2, . . . ,

where 𝑥0 is a point on R and ℎ𝑥 > 0 is a step of the space discretization. Using, for instance, linear interpolation, we construct the following algorithm on the basis of the layer method (5.4): (5.6)

{ 𝑣¯(𝑡𝑘 , 𝑥𝑗 ) =

𝑣¯(𝑡𝑁 , 𝑥) = 𝜑(𝑥), 1 ( 𝑣¯ 𝑡𝑘+1 , 𝑥𝑗 + ℎ [𝑏(𝑡𝑘 , 𝑥𝑗 ) − 𝛼(𝑡𝑘 , 𝑥𝑗 )𝛽(𝑡𝑘 , 𝑥𝑗 )] 2 ) +ℎ1/2 𝜎(𝑡𝑘 , 𝑥𝑗 ) + 𝛼(𝑡𝑘 , 𝑥𝑗 )Δ𝑘 𝑤

)} 1 ( 𝑣¯ 𝑡𝑘+1 , 𝑥𝑗 + ℎ [𝑏(𝑡𝑘 , 𝑥𝑗 ) − 𝛼(𝑡𝑘 , 𝑥𝑗 )𝛽(𝑡𝑘 , 𝑥𝑗 )] − ℎ1/2 𝜎(𝑡𝑘 , 𝑥𝑗 ) + 𝛼(𝑡𝑘 , 𝑥𝑗 )Δ𝑘 𝑤 2 × [1 + ℎ𝑐(𝑡𝑘 , 𝑥𝑗 ) + 𝛽(𝑡𝑘 , 𝑥𝑗 )Δ𝑘 𝑤] + ℎ𝑓 (𝑡𝑘 , 𝑥𝑗 ) + 𝛾(𝑡𝑘 , 𝑥𝑗 )Δ𝑘 𝑤, 𝑗 = 0, ±1, ±2, . . . , 𝑥𝑗+1 − 𝑥 𝑥 − 𝑥𝑗 𝑣¯(𝑡𝑘 , 𝑥) = (5.7) 𝑣¯(𝑡𝑘 , 𝑥𝑗 ) + 𝑣¯(𝑡𝑘 , 𝑥𝑗+1 ), 𝑥𝑗 ≤ 𝑥 ≤ 𝑥𝑗+1 , ℎ𝑥 ℎ𝑥 𝑘 = 𝑁 − 1, . . . , 1, 0. +

We use the same notation 𝑣¯ for the two different functions: one is defined by the layer algorithm (5.6)-(5.7) and the other is defined in Section 3.1 for (3.3), but this should not cause any confusion since we do not use the approximation (3.3) in the current section. If we need the solution of (2.1)-(2.2) for all points (𝑡𝑘 , 𝑥𝑖 ), we can use (5.6)-(5.7) to find 𝑣¯(𝑡𝑘 , 𝑥𝑗 ) layerwise. But if we need the solution at a particular point (𝑡𝑘 , 𝑥), the formula (3.10) is more convenient.


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Remark 5.1. Thanks to the probabilistic approach, we do not need any stability criteria for the layer methods in comparison with finite-difference schemes, on which the Lax-Richtmyer equivalence theorem imposes a requirement on the relation between the time step ℎ and the space step ℎ𝑥 . E.g., for 𝑏 = 0, 𝛼 = 0, 𝑐 = 0, 𝛽 = 0, and 𝜎 is a constant, the method (5.4) (we note that it does not need an interpolation in this case) coincides with the stable finite-difference scheme (see [10] and the references therein) 𝑣 (𝑡𝑘+1 , 𝑥𝑗 ) + 𝑣¯(𝑡𝑘+1 , 𝑥𝑗−1 ) 𝜎 2 𝑣¯(𝑡𝑘+1 , 𝑥𝑗+1 ) − 2¯ 𝑣¯(𝑡𝑘 , 𝑥𝑗 ) − 𝑣¯(𝑡𝑘+1 , 𝑥𝑗 ) = ℎ 2 ℎ2𝑥 Δ𝑘 𝑤 + 𝑓 (𝑡𝑘 , 𝑥𝑗 ) + 𝛾(𝑡𝑘 , 𝑥𝑗 ) ℎ √ with ℎ𝑥 = 𝜎 ℎ. In layer methods the suitable choice of points at which 𝑣¯ has to be evaluated is achieved automatically by taking into account the coefficient dependence on the space variables and a relationship between various terms (driving noise, diffusion, and advection) in an intrinsic manner. To prove the next convergence theorem, we need to impose additional assumptions on the considered problem to Assumptions 2.1-2.3 from Section 2. Assumption 5.1. We assume that the coefficients 𝑎, 𝑏, 𝛼𝑟 are uniformly bounded and that the function 𝜑 has bounded derivatives up to a sufficiently high order (i.e., it is globally Lipschitz). Theorem 5.1. Let Assumptions 2.1, 2.3, and 5.1 hold. The algorithm (5.6)-(5.7) with ℎ𝑥 = ϰℎ3/4 , ϰ > 0, satisfies the inequality for 𝑝 ≥ 1 : ( )1/(2𝑝) 2𝑝 ≤ 𝐾ℎ1/2 , (5.8) 𝐸 ∣¯ 𝑣 (𝑡𝑘 , 𝑥) − 𝑣(𝑡𝑘 , 𝑥)∣ where 𝐾 does not depend on 𝑥, ℎ, 𝑘, i.e., in particular, (5.6)-(5.7) is of mean-square order 1/2. For almost every trajectory 𝑤(⋅) and any 𝜀 > 0 the algorithm (5.6)-(5.7) with ℎ𝑥 = ϰℎ3/4 converges with order 1/2 − 𝜀, i.e., (5.9)

∣𝑣(𝑡𝑘 , 𝑥) − 𝑣¯(𝑡𝑘 , 𝑥)∣ ≤ 𝐶(𝜔)ℎ1/2−𝜀

𝑎.𝑠.,

where 𝐶 does not depend on 𝑥, ℎ, 𝑘. Proof. We first prove that ( )1/(2𝑝) 2𝑝 (5.10) 𝐸 ∣¯ 𝑣 (𝑡𝑘 , 𝑥𝑗 ) − 𝑣(𝑡𝑘 , 𝑥𝑗 )∣ ≤ 𝐾ℎ1/2 . In connection with the algorithm (5.6)-(5.7), we introduce the random sequence ˘ 𝑖 , 𝑌˘𝑖 , 𝑍˘𝑖 , 𝑖 = 𝑘, . . . , 𝑁, in the following way. We fix 𝑘 and put 𝑋 ˘ 𝑘 = 𝑥𝑗 , 𝑌˘𝑘 = 1, 𝑋 ˘ ˘ ˘ 𝑍𝑘 = 0 (to avoid confusion, we note that the index 𝑘 of 𝑋𝑘 , 𝑌𝑘 , 𝑍˘𝑘 means that ˘ 𝑘 , 𝑌˘𝑘 , 𝑍˘𝑘 belong to the 𝑘𝑡ℎ time layer, while the index 𝑗 of 𝑥𝑗 corresponds to the 𝑋 ˘ ± at ˘ 𝑖 , we introduce the auxiliary values 𝑋 space discretization (5.5)). Knowing 𝑋 𝑖+1 the (𝑖 + 1)-th step, 𝑖 = 𝑘, . . . , 𝑁 − 1: (5.11) [ ] ˘ ± := 𝑋 ˘ 𝑖 + ℎ 𝑏(𝑡𝑖 , 𝑋 ˘ 𝑖 ) − 𝛼(𝑡𝑖 , 𝑋 ˘ 𝑖 )𝛽(𝑡𝑖 , 𝑋 ˘ 𝑖 ) ± ℎ1/2 𝜎(𝑡𝑖 , 𝑋 ˘ 𝑖 ) + 𝛼(𝑡𝑖 , 𝑋 ˘ 𝑖 )Δ𝑖 𝑤. 𝑋 𝑖+1

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˘ 𝑖+1 for 𝑖 = 𝑘, . . . , 𝑁 − 2 is defined as the random variable distributed Then 𝑋 according to the law: ˘− ˘− ˘ 𝑖+1 = 𝑥𝑙 ) = 1 𝑥𝑙+1 − 𝑋𝑖+1 , 𝑃 (𝑋 ˘ 𝑖+1 = 𝑥𝑙+1 ) = 1 𝑋𝑖+1 − 𝑥𝑙 , 𝑃 (𝑋 2 ℎ𝑥 2 ℎ𝑥 + ˘+ ˘ 𝑥 𝑋 − 𝑋 𝑖+1 ˘ 𝑖+1 = 𝑥𝑚+1 ) = 1 𝑖+1 − 𝑥𝑚 , ˘ 𝑖+1 = 𝑥𝑚 ) = 1 𝑚+1 , 𝑃 (𝑋 𝑃 (𝑋 2 ℎ𝑥 2 ℎ𝑥 − ˘ ˘+ where 𝑥𝑙 , 𝑥𝑙+1 , 𝑥𝑚 , 𝑥𝑚+1 are such that 𝑥𝑙 ≤ 𝑋 𝑖+1 < 𝑥𝑙+1 , 𝑥𝑚 < 𝑋𝑖+1 ≤ 𝑥𝑚+1 ; ˘ 𝑁 is distributed as and 𝑋

𝑌˘𝑖+1 and 𝑍˘𝑖+1

˘𝑁 = 𝑋 ˘ − ) = 𝑃 (𝑋 ˘𝑁 = 𝑋 ˘ +) = 1 ; 𝑃 (𝑋 𝑁 𝑁 2 for 𝑖 = 𝑘, . . . , 𝑁 − 1 are defined in the following way: [ ] ˘ 𝑖 ) + 𝛽(𝑡𝑖 , 𝑋 ˘ 𝑖 )Δ𝑖 𝑤 , 𝑌˘𝑖+1 = 𝑌˘𝑖 1 + ℎ𝑐(𝑡𝑖 , 𝑋 ˘ 𝑖 )𝑌˘𝑖 + 𝛾(𝑡𝑖 , 𝑋 ˘ 𝑖 )𝑌˘𝑖 Δ𝑖 𝑤. 𝑍˘𝑖+1 = 𝑍˘𝑖 + ℎ𝑓 (𝑡𝑖 , 𝑋

It can be directly verified that

[ ] ˘ 𝑘+1 )𝑌˘𝑘+1 + 𝑍˘𝑘+1 . 𝑣¯(𝑡𝑘 , 𝑥𝑗 ) = 𝐸 𝑤 𝑣¯(𝑡𝑘+1 , 𝑋

(5.12)

˘ 𝑘+1 takes the values on the Using equalities of the type (3.20) and the fact that 𝑋 set of knots of the space discretization, we get [ ] ˘ 𝑘+2 )𝑌˘𝑘+2 + 𝑍˘𝑘+2 . 𝑣¯(𝑡𝑘 , 𝑥𝑗 ) = 𝐸 𝑤 𝑣¯(𝑡𝑘+2 , 𝑋 Carrying on in this way, we obtain that 𝑣¯(𝑡𝑘 , 𝑥𝑗 ) admits the following probabilistic representation: [ ] [ ] ˘ 𝑁 )𝑌˘𝑁 + 𝑍˘𝑁 = 𝐸 𝑤 𝜑(𝑋 ˘ 𝑁 )𝑌˘𝑁 + 𝑍˘𝑁 . (5.13) 𝑣¯(𝑡𝑘 , 𝑥𝑗 ) = 𝐸 𝑤 𝑣¯(𝑡𝑁 , 𝑋 ity

Now we shall act analogously to the proof of Theorem 3.2 and prove the inequal)1/(2𝑝) ( ≤ 𝐾ℎ1/2 . 𝐸 ∣¯ 𝑣 (𝑡𝑘 , 𝑥𝑗 ) − 𝑣˜(𝑡𝑘 , 𝑥𝑗 )∣2𝑝

(5.14)

Then the inequality (5.10) follows from Theorem 3.2 (see (3.11)). We write the difference 𝐷𝑁 := 𝑣¯(𝑡𝑘 , 𝑥𝑗 )− 𝑣˜(𝑡𝑘 , 𝑥𝑗 ) in the form (see (3.15)-(3.16)): (5.15)

] [ [ ] ˘ 𝑡 ,𝑥 (𝑡𝑁 )) 𝑌˘𝑡 ,𝑥 ,1 (𝑡𝑁 ) + 𝑍˘𝑡 ,𝑥 ,1,0 (𝑡𝑁 ) − 𝐸 𝑤 𝜑(𝑋 ˜ 𝑁 )𝑌˜𝑁 + 𝑍˜𝑁 𝐷𝑁 = 𝐸 𝑤 𝜑(𝑋 𝑘 𝑗 𝑘 𝑗 𝑘 𝑗 = 𝐸𝑤

𝑁 −1 ∑

˘ 𝑖 , 𝑌˘𝑖 , 𝑍˘𝑖 ), 𝜌𝑖 ( 𝑋

𝑖=0

where (5.16)

˘ 𝑖 , 𝑌˘𝑖 , 𝑍˘𝑖 ) = 𝐸 𝑤,𝑋˘ 𝑖 ,𝑌˘𝑖 ,𝑍˘𝑖 [˜ ˘ 𝑖+1 )𝑌˘𝑖+1 + 𝑍˘𝑖+1 𝜌𝑖 ( 𝑋 𝑣 (𝑡𝑖+1 , 𝑋 ˜ ˘ (𝑡𝑖+1 ))𝑌˜ ˘ ˘ (𝑡𝑖+1 ) − 𝑍˜ ˘ ˘ ˘ (𝑡𝑖+1 )]. −˜ 𝑣 (𝑡𝑖+1 , 𝑋 𝑡𝑖 ,𝑋𝑖

𝑡𝑖 ,𝑋𝑖 ,𝑌𝑖

𝑡𝑖 ,𝑋𝑖 ,𝑌𝑖 ,𝑍𝑖

One can prove that the function 𝑣˜(𝑡𝑖 , 𝑥) from (3.10) is sufficiently smooth in 𝑥 and satisfies (together with its derivatives) the same inequality as the function 𝑣(𝑡, 𝑥) (see the inequality (2.5)).


2095

˘ 𝑖 , 𝑌˘𝑖 , 𝑍˘𝑖 ). Since 𝑌˘𝑖+1 = 𝑌˜ ˘ ˘ (𝑡𝑖+1 ), Our next step is to get an estimate for 𝜌𝑖 (𝑋 𝑡𝑖 ,𝑋𝑖 ,𝑌𝑖 𝑍˘𝑖+1 = 𝑍˜𝑡𝑖 ,𝑋˘ 𝑖 ,𝑌˘𝑖 ,𝑍˘𝑖 (𝑡𝑖+1 ), we have ˘ 𝑖 , 𝑌˘𝑖 , 𝑍˘𝑖 ) = 𝜌𝑖 (𝑋 ˘ 𝑖 , 𝑌˘𝑖 ) = 𝐸 𝑤,𝑋˘ 𝑖 ,𝑌˘𝑖 𝑌˘𝑖+1 [˜ ˘ 𝑖+1 ) − 𝑣˜(𝑡𝑖+1 , 𝑋 ˜ ˘ (𝑡𝑖+1 ))] 𝑣 (𝑡𝑖+1 , 𝑋 𝜌𝑖 ( 𝑋 𝑡𝑖 ,𝑋𝑖 ˘ ˘ ˘ 𝑖+1 ) − 𝑣˜(𝑡𝑖+1 , 𝑋 ˜ ˘ (𝑡𝑖+1 ))]. = 𝑌˘𝑖+1 𝐸 𝑤,𝑋𝑖 ,𝑌𝑖 [˜ 𝑣 (𝑡𝑖+1 , 𝑋 𝑡𝑖 ,𝑋𝑖

˘ 𝑖+1 ) with respect to powers of Δ ˘ 𝑖 := 𝑋 ˘ 𝑖+1 − 𝑋 ˘ 𝑖 in a We expand 𝑣˜(𝑡𝑖+1 , 𝑋 ˘ borhood of 𝑋𝑖 and with the Lagrange remainder term containing terms ˜ ˘ (𝑡𝑖+1 )) with respect to the der four, and we similarly expand 𝑣˜(𝑡𝑖+1 , 𝑋 𝑡𝑖 ,𝑋𝑖 ˜ ˘ (𝑡𝑖+1 ) − 𝑋 ˘ 𝑖 . One can verify that for 𝑝 ≥ 1: 𝑋

neighof or˜ 𝑖 := Δ

𝑡𝑖 ,𝑋𝑖

˘

˘

˘

˘

˘𝑖 − Δ ˜ 𝑖 ] = 𝐸 𝑤,𝑋𝑖 ,𝑌𝑖 [𝑋 ˘ 𝑖+1 − 𝑋 ˜ ˘ (𝑡𝑖+1 )] 𝐸 𝑤,𝑋𝑖 ,𝑌𝑖 [Δ 𝑡𝑖 ,𝑋𝑖 ˘ ˘ ˜ ˘ (𝑡𝑖+1 )] = 0, = 𝐸 𝑤,𝑋𝑖 [𝑋 𝑖+1 − 𝑋𝑡𝑖 ,𝑋 𝑖 )2𝑝 ( ˘ 𝑖 ,𝑌˘𝑖 ˘ 𝑟 𝑤,𝑋 𝑟 4𝑝 ˜𝑖] [Δ𝑖 − Δ = 𝑂(ℎ𝑥 ), 𝑟 = 2, 3, 𝐸 𝐸

√ and for ℎ𝑥 ≤ ϰ ℎ, ) ( ) ( ˘ ˘ ˘ 4 2𝑝 ˘ ˘ ˜ 4 2𝑝 = 𝑂(ℎ4𝑝 ), 𝐸 𝐸 𝑤,𝑋𝑖 ,𝑌𝑖 [Δ = 𝑂(ℎ4𝑝 ). 𝐸 𝐸 𝑤,𝑋𝑖 ,𝑌𝑖 [Δ 𝑖] 𝑖] Then, taking into account ℎ𝑥 = ϰℎ3/4 , it is not difficult to prove that 2𝑝 ˘ 𝑖 , 𝑌˘𝑖 , 𝑍˘𝑖 ) ≤ 𝐾ℎ3𝑝 . 𝐸 𝜌𝑖 (𝑋

From here, using the same arguments as in the proof of Theorem 3.2 (see (3.22)), we obtain (5.14) and, consequently, (5.10). Due to the smoothness of 𝑣(𝑡, 𝑥) in 𝑥, we have 𝑥𝑗+1 − 𝑥 𝑥 − 𝑥𝑗 𝑣(𝑡𝑘 , 𝑥𝑗 ) + 𝑣(𝑡𝑘 , 𝑥𝑗+1 ) + 𝑂(ℎ2𝑥 ), 𝑥𝑗 ≤ 𝑥 ≤ 𝑥𝑗+1 , 𝑣(𝑡𝑘 , 𝑥) = ℎ𝑥 ℎ𝑥 where 𝐸∣𝑂(ℎ2𝑥 )∣2𝑝 ≤ 𝐾ℎ4𝑝 𝑥 . Then 𝑥𝑗+1 − 𝑥 𝑣¯(𝑡𝑘 , 𝑥) − 𝑣(𝑡𝑘 , 𝑥) = [¯ 𝑣 (𝑡𝑘 , 𝑥𝑗 ) − 𝑣(𝑡𝑘 , 𝑥𝑗 )] ℎ𝑥 𝑥 − 𝑥𝑗 + [¯ 𝑣 (𝑡𝑘 , 𝑥𝑗+1 ) − 𝑣(𝑡𝑘 , 𝑥𝑗+1 )] + 𝑂(ℎ2𝑥 ), ℎ𝑥 and, using (5.10) and that ℎ𝑥 = ϰℎ3/4 , we obtain ( 𝐸∣¯ 𝑣 (𝑡𝑘 , 𝑥) − 𝑣(𝑡𝑘 , 𝑥)∣2𝑝 ≤ 𝐾⋅ 𝐸∣¯ 𝑣 (𝑡𝑘 , 𝑥𝑗 ) − 𝑣(𝑡𝑘 , 𝑥𝑗 )∣2𝑝

) +𝐸∣¯ 𝑣 (𝑡𝑘 , 𝑥𝑗+1 ) − 𝑣(𝑡𝑘 , 𝑥𝑗+1 )∣2𝑝 + 𝐾ℎ3𝑝 ≤ 𝐾ℎ𝑝 ,

whence (5.8) follows. The inequality (5.9) is proved as its analogue in Theorem 3.1. □ We note that for some particular SPDEs (see Remark 3.1) the algorithm (5.6)(5.7) with ℎ𝑥 = ϰℎ converges with the weak order 1 − 𝜀. Remark 5.2. The generalization to the multidimensional SPDE, 𝑑 > 1, 𝑞 ≥ 1, is straightforward (see it in the case of deterministic PDEs in [21, 23]). We also note that other types of interpolation can be exploited here to construct layer algorithms [21, 23].

2096


6. Layer methods for semilinear SPDEs In this section, we generalize layer methods for the linear SPDEs considered in Section 5 to the case of semilinear SPDEs (see layer methods for semilinear deterministic PDEs in [21, 23] and for quasilinear deterministic PDEs in [24]). As in the previous section, we restrict ourselves to the one-dimensional (𝑑 = 1) and one noise (𝑞 = 1) case for simplicity in writing only. There is no difficulty in generalizing the methods of this section to the multi-dimensional multi-noise case (see how this is done in the case of deterministic PDEs in, e.g., [21, 23]). We consider the Cauchy problem for the backward semilinear SPDE: (6.1)

(6.2)

− 𝑑𝑣 = [𝐿𝑣 + 𝑓 (𝑡, 𝑥, 𝑣)] 𝑑𝑡 ] [ ∂ + 𝛼(𝑡, 𝑥, 𝑣) 𝑣 + 𝛾(𝑡, 𝑥, 𝑣) ∗ 𝑑𝑤(𝑡), (𝑡, 𝑥) ∈ [𝑇0 , 𝑇 ) × R, ∂𝑥 𝑣(𝑇, 𝑥) = 𝜑(𝑥), 𝑥 ∈ R,

where (6.3)

1 ∂2 ∂ 𝑎(𝑡, 𝑥, 𝑣(𝑡, 𝑥)) 2 𝑣(𝑡, 𝑥) + 𝑏(𝑡, 𝑥, 𝑣(𝑡, 𝑥)) 𝑣(𝑡, 𝑥), 2 ∂𝑥 ∂𝑥 𝑓 (𝑡, 𝑥, 𝑣) := 𝑓0 (𝑡, 𝑥)𝑣 + 𝑓1 (𝑡, 𝑥, 𝑣), 𝛾(𝑡, 𝑥, 𝑣) := 𝛾0 (𝑡, 𝑥)𝑣 + 𝛾1 (𝑡, 𝑥, 𝑣). 𝐿𝑣(𝑡, 𝑥) :=

Assumption 6.1. We assume that the coefficients 𝑎(𝑡, 𝑥, 𝑣), 𝑏(𝑡, 𝑥, 𝑣), 𝑓0 (𝑡, 𝑥), 𝑓1 (𝑡, 𝑥, 𝑣), 𝛼(𝑡, 𝑥, 𝑣), 𝛾0 (𝑡, 𝑥), and 𝛾1 (𝑡, 𝑥, 𝑣) and the function 𝜑(𝑥) are bounded and have bounded derivatives up to some order and that the coercivity condition is satisfied, i.e., 𝜎 2 (𝑡, 𝑥, 𝑣) := 𝑎(𝑡, 𝑥, 𝑣) − 𝛼2 (𝑡, 𝑥, 𝑣) ≥ 0 for all 𝑡, 𝑥, 𝑣. We also suppose that the problem (6.1)-(6.2) has the classical solution 𝑣(𝑡, 𝑥), which has derivatives in 𝑥 up to a sufficiently high order, and that the solution and its spatial derivatives have bounded moments up to some order. We note that under Assumption 6.1 the existence of the classical solution is proved in the case 𝑎(𝑡, 𝑥, 𝑣) = 𝑎(𝑡, 𝑥), 𝑏(𝑡, 𝑥, 𝑣) = 𝑏(𝑡, 𝑥), 𝛼(𝑡, 𝑥, 𝑣) = 𝛼(𝑡, 𝑥) in [14] and in the case 𝑎(𝑡, 𝑥, 𝑣) = 𝑎(𝑡, 𝑥) in [28, 30]. Formally generalizing the one-step approximation (5.3), we get the one-step approximation for the semilinear problem (6.1)-(6.2): 1 ( (6.4) 𝑣˘(𝑡𝑘 , 𝑥) = 𝑣 𝑡𝑘+1 , 𝑥 + ℎ𝑏(𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) 2 ) +ℎ1/2 𝜎(𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) + 𝛼(𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥))Δ𝑘 𝑤 1 ( + 𝑣 𝑡𝑘+1 , 𝑥 + ℎ𝑏(𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) 2 ) −ℎ1/2 𝜎(𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) + 𝛼(𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥))Δ𝑘 𝑤 + ℎ𝑓 (𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) + 𝛾(𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥))Δ𝑘 𝑤. In the rest of this section we use the same letter 𝐾 for various deterministic constants and 𝐶 = 𝐶(𝜔) for various positive random variables. Lemma 6.1. Let Assumptions 6.1 hold. The error 𝜌 = 𝜌(𝑡𝑘 ) = 𝑣˘(𝑡𝑘 , 𝑥) − 𝑣(𝑡𝑘 , 𝑥) of the one-step approximation (6.4) is estimated as (6.5)

𝑡

𝑡

∣𝐸(𝜌∣ℱ𝑇𝑘+1 )∣ ≤ 𝐶(𝜔)ℎ2 , ∣𝐸(Δ𝑘 𝑤𝜌∣ℱ𝑇𝑘+1 )∣ ≤ 𝐶(𝜔)ℎ2 ,


(6.6)

2097

( )1/(2𝑝) 𝐸 [𝜌]2𝑝 ≤ 𝐾ℎ, 𝑝 ≥ 1,

where 𝐶(𝜔) and 𝐾 do not depend on ℎ, 𝑘, 𝑥 and 𝐸𝐶 2 < ∞. In the case of additive noise, 𝛼 = 0 and 𝛾(𝑡, 𝑥, 𝑣) = 𝛾(𝑡, 𝑥) (i.e., 𝛾0 (𝑡, 𝑥) = 0, 𝛾1 (𝑡, 𝑥, 𝑣) = 𝛾(𝑡, 𝑥)), the error 𝜌 satisfies (6.5) and )1/(2𝑝) ( (6.7) 𝐸 [𝜌]2𝑝 ≤ 𝐾ℎ3/2 , 𝑝 ≥ 1. Proof. We note that the increments Δ𝑘 𝑤 and the values 𝑣(𝑡𝑘+1 , 𝑥), ∂ 𝑗 𝑣(𝑡𝑘+1 , 𝑥)/∂𝑥𝑗 are independent. Using the assumptions, we first expand the right-hand side of (6.4) at (𝑡𝑘+1 , 𝑥) with the Lagrange remainder containing fourth-order spatial derivatives and then obtain (6.8) ∂𝑣 (𝑡𝑘+1 , 𝑥) 𝑣˘(𝑡𝑘 , 𝑥) = 𝑣(𝑡𝑘+1 , 𝑥) + [𝛼(𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥))Δ𝑘 𝑤 + ℎ𝑏(𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥))] ∂𝑥 ] ∂2𝑣 1[ 2 + ℎ𝜎 (𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) + 𝛼2 (𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) (Δ𝑘 𝑤)2 (𝑡𝑘+1 , 𝑥) 2 ∂𝑥2 + ℎ𝑓 (𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) + 𝛾(𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥))Δ𝑘 𝑤 + 𝑟1 (𝑡𝑘+1 , 𝑥), where the remainder 𝑟1 (𝑡𝑘+1 , 𝑥) satisfies the inequalities (in the general case) (6.9)

) ) ( ( 𝑡 𝑡 ∣𝐸 𝑟1 (𝑡𝑘+1 , 𝑥)∣ℱ𝑇𝑘+1 ∣ ≤ 𝐶(𝜔)ℎ2 , ∣𝐸 Δ𝑘 𝑤𝑟1 (𝑡𝑘+1 , 𝑥)∣ℱ𝑇𝑘+1 ∣ ≤ 𝐶(𝜔)ℎ2 , ( )1/(2𝑝) 𝐸 [𝑟1 (𝑡𝑘+1 , 𝑥)]2𝑝 ≤ 𝐾ℎ3/2 . Introduce the operators: ∂ ∂ ℒ𝑥 𝜓(𝑡, 𝑧) := 𝜓(𝑡, 𝑧) + [𝐿𝑣(𝑡, 𝑥) + 𝑓 (𝑡, 𝑥, 𝑣(𝑡, 𝑥))] 𝜓(𝑡, 𝑧) ∂𝑡 ∂𝑧 ]2 2 [ 1 ∂ ∂ + 𝜓(𝑡, 𝑧), 𝛼(𝑡, 𝑥, 𝑣(𝑡, 𝑥)) 𝑣 + 𝛾(𝑡, 𝑥, 𝑣(𝑡, 𝑥)) 2 ∂𝑥 ∂𝑧 2 ] [ ∂ ∂ 𝜓(𝑡, 𝑧). Λ𝑥 𝜓(𝑡, 𝑧) := 𝛼(𝑡, 𝑥, 𝑣(𝑡, 𝑥)) 𝑣 + 𝛾(𝑡, 𝑥, 𝑣(𝑡, 𝑥)) ∂𝑥 ∂𝑧

Due to the backward Ito formula [32, 28], we have for a smooth 𝜓(𝑡, 𝑧) and 𝑡 ≤ 𝑡𝑘+1 : ∫ 𝑡𝑘+1 𝜓(𝑡, 𝑣(𝑡, 𝑥)) = 𝜓(𝑡𝑘+1 , 𝑣(𝑡𝑘+1 , 𝑥)) + (6.10) ℒ𝑥 𝜓(𝑠, 𝑣(𝑠, 𝑥))𝑑𝑠 𝑡 ∫ 𝑡𝑘+1 Λ𝑥 𝜓(𝑠, 𝑣(𝑠, 𝑥)) ∗ 𝑑𝑤(𝑠). + 𝑡

We write the solution 𝑣(𝑡, 𝑥), 𝑇0 ≤ 𝑡 ≤ 𝑡𝑘+1 , of (6.1)-(6.2) as ∫ 𝑡𝑘+1 (6.11) [𝐿𝑣(𝑠, 𝑥) + 𝑓 (𝑠, 𝑥, 𝑣(𝑠, 𝑥))] 𝑑𝑠 𝑣(𝑡, 𝑥) = 𝑣(𝑡𝑘+1 , 𝑥) + 𝑡 ] ∫ 𝑡𝑘+1 [ ∂𝑣 + 𝛼(𝑠, 𝑥, 𝑣(𝑠, 𝑥)) (𝑠, 𝑥) + 𝛾(𝑠, 𝑥, 𝑣(𝑠, 𝑥)) ∗ 𝑑𝑤(𝑠) ∂𝑥 𝑡 and, in particular, ∫ 𝑡𝑘+1 𝑣(𝑡𝑘 , 𝑥) = 𝑣(𝑡𝑘+1 , 𝑥) + (6.12) [𝐿𝑣(𝑠, 𝑥) + 𝑓 (𝑠, 𝑥, 𝑣(𝑠, 𝑥))] 𝑑𝑠 ∫ +

𝑡𝑘+1

𝑡𝑘

[

𝑡𝑘

𝛼(𝑠, 𝑥, 𝑣(𝑠, 𝑥))

] ∂𝑣 (𝑠, 𝑥) + 𝛾(𝑠, 𝑥, 𝑣(𝑠, 𝑥)) ∗ 𝑑𝑤(𝑠). ∂𝑥

2098


We substitute the expression for 𝑣(𝑠, 𝑥) from (6.11) in the parts of the integrand in (6.12): (6.13)

∫

𝑠,𝑥

𝑡𝑘+1

𝐿𝑠,𝑥 [𝐿𝑣(𝑠′ , 𝑥) + 𝑓 (𝑠′ , 𝑥, 𝑣(𝑠′ , 𝑥))] 𝑑𝑠′ 𝐿𝑣(𝑠, 𝑥) = 𝐿 𝑣(𝑡𝑘+1 , 𝑥) + 𝑠 ] [ ∫ 𝑡𝑘+1 ∂𝑣 ′ 𝑠,𝑥 ′ ′ ′ ′ 𝐿 𝛼(𝑠 , 𝑥, 𝑣(𝑠 , 𝑥)) (𝑠 , 𝑥) + 𝛾(𝑠 , 𝑥, 𝑣(𝑠 , 𝑥)) ∗ 𝑑𝑤(𝑠′ ), + ∂𝑥 𝑠 ∫ 𝑡𝑘+1 ∂𝑣 ∂ ∂𝑣 (𝑠, 𝑥) = (𝑡𝑘+1 , 𝑥) + [𝐿𝑣(𝑠′ , 𝑥) + 𝑓 (𝑠′ , 𝑥, 𝑣(𝑠′ , 𝑥))] 𝑑𝑠′ ∂𝑥 ∂𝑥 ∂𝑥 𝑠 ] [ ∫ 𝑡𝑘+1 ∂ ∂𝑣 + 𝛼(𝑠′ , 𝑥, 𝑣(𝑠′ , 𝑥)) (𝑠′ , 𝑥) + 𝛾(𝑠′ , 𝑥, 𝑣(𝑠′ , 𝑥)) ∗ 𝑑𝑤(𝑠′ ), ∂𝑥 ∂𝑥 𝑠 where 𝐿𝑠,𝑥 𝜓(𝑡, 𝑧) :=

∂2 1 ∂ 𝑎(𝑠, 𝑥, 𝑣(𝑠, 𝑥)) 2 𝜓(𝑡, 𝑧) + 𝑏(𝑠, 𝑥, 𝑣(𝑠, 𝑥)) 𝜓(𝑡, 𝑧). 2 ∂𝑧 ∂𝑧

Now we apply the Ito formula (6.10) to 𝑓, 𝛼 and 𝛾 in (6.12) and to the coefficients 𝑎 and 𝑏 appearing in 𝐿𝑠,𝑥 𝑣(𝑡𝑘+1 , 𝑥) in (6.13). Then taking into account the assumptions and properties of Ito integrals, we obtain [

(6.14)

1 ∂2 𝑎(𝑡𝑘+1 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) 2 2 ∂𝑥 ] ∂ +𝑏(𝑡𝑘+1 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) 𝑣(𝑡𝑘+1 , 𝑥) ∂𝑥 + ℎ𝑓 (𝑡𝑘+1 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) [ ∂𝑣 + 𝛼(𝑡𝑘+1 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) (𝑡𝑘+1 , 𝑥) ∂𝑥 ] + 𝛾(𝑡𝑘+1 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) Δ𝑘 𝑤 + 𝑟2 (𝑡𝑘+1 , 𝑥),

𝑣(𝑡𝑘 , 𝑥) = 𝑣(𝑡𝑘+1 , 𝑥) + ℎ

where the remainder 𝑟2 (𝑡𝑘+1 , 𝑥) satisfies the inequalities of the form (6.5)-(6.6) in the general case and of the form (6.9) in the case 𝛼 = 0, 𝛾(𝑡, 𝑥, 𝑣) = 𝛾(𝑡, 𝑥). Furthermore, in (6.14) we expand the coefficients at (𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) and the new remainder has the same properties as 𝑟2 . The obtained result is subtracted from (6.8) and we get (taking into account that 𝑎 = 𝜎 2 + 𝛼2 ) (6.15) [ ] 1 ∂2𝑣 (𝑡𝑘+1 , 𝑥)𝛼2 (𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) (Δ𝑘 𝑤)2 − ℎ +𝑟(𝑡𝑘+1 , 𝑥), 𝜌 = 𝑣˘(𝑡𝑘 , 𝑥)−𝑣(𝑡𝑘 , 𝑥) = 2 2 ∂𝑥 where the remainder 𝑟(𝑡𝑘+1 , 𝑥) has the same properties as 𝑟2 . Lemma 6.1 evidently follows from (6.15). □


2099

Based on the one-step approximation (6.4), we obtain the layer method: (6.16)

𝑣¯(𝑡𝑁 , 𝑥) = 𝜑(𝑥), ( 1 𝑣¯(𝑡𝑘 , 𝑥) = 𝑣¯ 𝑡𝑘+1 , 𝑥 + ℎ𝑏(𝑡𝑘 , 𝑥, 𝑣¯(𝑡𝑘+1 , 𝑥)) 2 +ℎ1/2 𝜎(𝑡𝑘 , 𝑥, 𝑣¯(𝑡𝑘+1 , 𝑥)) + 𝛼(𝑡𝑘 , 𝑥, 𝑣¯(𝑡𝑘+1 , 𝑥))Δ𝑘 𝑤

1 ( + 𝑣¯ 𝑡𝑘+1 , 𝑥 + ℎ𝑏(𝑡𝑘 , 𝑥, 𝑣¯(𝑡𝑘+1 , 𝑥)) 2

−ℎ1/2 𝜎(𝑡𝑘 , 𝑥, 𝑣¯(𝑡𝑘+1 , 𝑥)) + 𝛼(𝑡𝑘 , 𝑥, 𝑣¯(𝑡𝑘+1 , 𝑥))Δ𝑘 𝑤

)

)

+ℎ𝑓 (𝑡𝑘 , 𝑥, 𝑣¯(𝑡𝑘+1 , 𝑥)) + 𝛾(𝑡𝑘 , 𝑥, 𝑣¯(𝑡𝑘+1 , 𝑥))Δ𝑘 𝑤, 𝑘 = 𝑁 − 1, . . . , 1, 0. Let us prove the following technical lemma, which will be exploited in proving the convergence Theorem 6.1. Lemma 6.2. Let Assumptions 6.1 hold. Then (6.17)

𝑅(𝑡𝑘 , 𝑥) := 𝑣¯(𝑡𝑘 , 𝑥) − 𝑣(𝑡𝑘 , 𝑥) 1 1 ¯ ℎ1/2 + 𝛼Δ ¯ 𝑘 𝑤) + 𝑅(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ − 𝜎 ¯ ℎ1/2 + 𝛼Δ ¯ 𝑘 𝑤) = 𝑅(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ + 𝜎 2 2 ∂𝑣 (𝑡𝑘+1 , 𝑥)(Δ𝑏ℎ + Δ𝛼Δ𝑘 𝑤) + 𝑟 + 𝜌, +Δ𝑓 ℎ + Δ𝛾Δ𝑘 𝑤 + ∂𝑥 where 𝜌 is the one-step error as in Lemma 6.1 and ∣ 𝑟 ∣ ≤ 𝐶 ∣ 𝑅(𝑡𝑘+1 , 𝑥) ∣ (ℎ + Δ2𝑘 𝑤). Proof. From (6.16) and due to the equality 𝑣¯(𝑡𝑘+1 , ⋅) = 𝑣(𝑡𝑘+1 , ⋅) + 𝑅(𝑡𝑘+1 , ⋅), we get (6.18)

1 1 𝑣(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ + 𝜎 ¯ ℎ1/2 + 𝛼Δ ¯ 𝑘 𝑤) + 𝑣(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ − 𝜎 ¯ ℎ1/2 + 𝛼Δ ¯ 𝑘 𝑤) 2 2 1 1 + 𝑅(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ + 𝜎 ¯ ℎ1/2 + 𝛼Δ ¯ 𝑘 𝑤) + 𝑅(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ − 𝜎 ¯ ℎ1/2 + 𝛼Δ ¯ 𝑘 𝑤) 2 2 +𝑓¯ℎ + 𝛾¯ Δ𝑘 𝑤, 𝑘 = 𝑁 − 1, . . . , 1, 0,

𝑣¯(𝑡𝑘 , 𝑥) =

where ¯𝑏, 𝜎 ¯ , 𝛼, ¯ 𝑓¯, 𝛾¯ are the coefficients 𝑏(𝑡, 𝑥, 𝑣), . . . , 𝛾(𝑡, 𝑥, 𝑣) calculated at 𝑡 = 𝑡𝑘 , 𝑥 = 𝑥, 𝑣 = 𝑣¯(𝑡𝑘+1 , 𝑥) = 𝑣(𝑡𝑘+1 , 𝑥) + 𝑅(𝑡𝑘+1 , 𝑥). We have (6.19) ¯𝑏 = 𝑏(𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥) + 𝑅(𝑡𝑘+1 , 𝑥)) = 𝑏(𝑡𝑘 , 𝑥, 𝑣(𝑡𝑘+1 , 𝑥)) + Δ𝑏 := 𝑏 + Δ𝑏, where Δ𝑏 satisfies the inequality ∣ Δ𝑏 ∣ ≤ 𝐾 ∣ 𝑅(𝑡𝑘+1 , 𝑥) ∣

(6.20)

with a deterministic constant 𝐾. Due to the uniform boundedness of 𝑏, we also get ∣ Δ𝑏 ∣2 ≤ ∣ Δ𝑏 ∣ ×𝐾 ∣ 𝑅(𝑡𝑘+1 , 𝑥) ∣ ≤ 𝐾 ∣ 𝑅(𝑡𝑘+1 , 𝑥) ∣ .

(6.21) Analogously, (6.22)

𝜎 ¯ = 𝜎 + Δ𝜎, ∣ Δ𝜎 ∣ ≤ 𝐾 ∣ 𝑅(𝑡𝑘+1 , 𝑥) ∣, ∣ Δ𝜎 ∣2 ≤ 𝐾 ∣ 𝑅(𝑡𝑘+1 , 𝑥) ∣, 𝛼 ¯ = 𝛼 + Δ𝛼, ∣ Δ𝛼 ∣ ≤ 𝐾 ∣ 𝑅(𝑡𝑘+1 , 𝑥) ∣, ∣ Δ𝛼 ∣2 ≤ 𝐾 ∣ 𝑅(𝑡𝑘+1 , 𝑥) ∣ .

2100


For Δ𝑓 and Δ𝛾, we have (6.23)

∣ Δ𝑓 ∣ ≤ 𝐾 ∣ 𝑅(𝑡𝑘+1 , 𝑥) ∣, ∣ Δ𝛾 ∣ ≤ 𝐾 ∣ 𝑅(𝑡𝑘+1 , 𝑥) ∣ .

It is not difficult to see that (6.20)-(6.22) imply the equalities ¯ ℎ1/2 + 𝛼 ¯ Δ𝑘 𝑤) = 𝑣(𝑡𝑘+1 , 𝑥 + 𝑏ℎ ± 𝜎ℎ1/2 + 𝛼Δ𝑘 𝑤) 𝑣(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ ± 𝜎

(6.24) +

∂𝑣 (𝑡𝑘+1 , 𝑥 + 𝑏ℎ ± 𝜎ℎ1/2 + 𝛼Δ𝑘 𝑤)(Δ𝑏ℎ ± Δ𝜎ℎ1/2 + Δ𝛼Δ𝑘 𝑤) + 𝑟, ∂𝑥

where (6.25)

∣ 𝑟 ∣ ≤ 𝐶 ∣ Δ𝑏ℎ ± Δ𝜎ℎ1/2 + Δ𝛼Δ𝑘 𝑤 ∣2 ≤ 𝐶 ∣ 𝑅(𝑡𝑘+1 , 𝑥) ∣ (ℎ + Δ2𝑘 𝑤),

and 𝐶 is a random variable with uniformly bounded second moment 𝐸𝐶 2 . Furthermore, ∂𝑣 ∂𝑣 (𝑡𝑘+1 , 𝑥+𝑏ℎ±𝜎ℎ1/2 +𝛼Δ𝑘 𝑤) = (𝑡𝑘+1 , 𝑥)+𝐶ℎ1/2 +𝐶Δ𝑘 𝑤, 𝐸𝐶 2 < ∞. ∂𝑥 ∂𝑥 Substituting (6.26) in (6.24) and then the new (6.24) in (6.18) and taking into account (6.20)-(6.22) with (6.25), we obtain (6.26)

(6.27)

1 1 𝑣(𝑡𝑘+1 , 𝑥 + 𝑏ℎ + 𝜎ℎ1/2 + 𝛼Δ𝑘 𝑤) + 𝑣(𝑡𝑘+1 , 𝑥 + 𝑏ℎ − 𝜎ℎ1/2 + 𝛼Δ𝑘 𝑤) 2 2 ∂𝑣 (𝑡𝑘+1 , 𝑥)(Δ𝑏ℎ + Δ𝛼Δ𝑘 𝑤) +𝑓 ℎ + 𝛾Δ𝑘 𝑤 + Δ𝑓 ℎ + Δ𝛾Δ𝑘 𝑤 + ∂𝑥 1 1 + 𝑅(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ + 𝜎 ¯ ℎ1/2 + 𝛼Δ ¯ 𝑘 𝑤) + 𝑅(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ − 𝜎 ¯ ℎ1/2 + 𝛼Δ ¯ 𝑘 𝑤) + 𝑟 2 2 1 1 = 𝑣˘(𝑡𝑘 , 𝑥)+ 𝑅(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ + 𝜎 ¯ ℎ1/2 + 𝛼Δ ¯ 𝑘 𝑤) + 𝑅(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ − 𝜎 ¯ ℎ1/2 + 𝛼Δ ¯ 𝑘 𝑤) 2 2 ∂𝑣 (𝑡𝑘+1 , 𝑥)(Δ𝑏ℎ + Δ𝛼Δ𝑘 𝑤) + 𝑟, +Δ𝑓 ℎ + Δ𝛾Δ𝑘 𝑤 + ∂𝑥 where the new 𝑟 also satisfies 𝑣¯(𝑡𝑘 , 𝑥) =

∣ 𝑟 ∣ ≤ 𝐶 ∣ 𝑅(𝑡𝑘+1 , 𝑥) ∣ (ℎ + Δ2𝑘 𝑤). At last using Lemma 6.1, we obtain 𝑅(𝑡𝑘 , 𝑥) = 𝑣¯(𝑡𝑘 , 𝑥) − 𝑣(𝑡𝑘 , 𝑥) = 𝑣˘(𝑡𝑘 , 𝑥) − 𝑣(𝑡𝑘 , 𝑥) 1 1 ¯ ℎ1/2 + 𝛼Δ ¯ 𝑘 𝑤) + 𝑅(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ − 𝜎 ¯ ℎ1/2 + 𝛼Δ ¯ 𝑘 𝑤) + 𝑅(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ + 𝜎 2 2 ∂𝑣 (𝑡𝑘+1 , 𝑥)(Δ𝑏ℎ + Δ𝛼Δ𝑘 𝑤) + 𝑟 +Δ𝑓 ℎ + Δ𝛾Δ𝑘 𝑤 + ∂𝑥 1 1 ¯ ℎ1/2 + 𝛼Δ ¯ 𝑘 𝑤) + 𝑅(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ − 𝜎 ¯ ℎ1/2 + 𝛼Δ ¯ 𝑘 𝑤) = 𝑅(𝑡𝑘+1 , 𝑥 + ¯𝑏ℎ + 𝜎 2 2 ∂𝑣 (𝑡𝑘+1 , 𝑥)(Δ𝑏ℎ + Δ𝛼Δ𝑘 𝑤) + 𝑟 + 𝜌, +Δ𝑓 ℎ + Δ𝛾Δ𝑘 𝑤 + ∂𝑥 whence (6.17) follows. (6.28)

□

Due to Lemmas 6.1 and 6.2, one can expect that the method (6.16) converges with the mean-square order 1/2. But we have not succeeded in proving the corresponding convergence theorem in the general case. Here we restrict ourselves to proving the following theorem.


2101

Theorem 6.1. Let Assumptions 6.1 hold. Let 𝛼 = 0 and 𝑏 and 𝜎 be independent of 𝑣. Then (𝐸(¯ 𝑣(𝑡𝑘 , 𝑥) − 𝑣(𝑡𝑘 , 𝑥))2 )1/2 ≤ 𝐾ℎ1/2 ,

(6.29)

where 𝐾 does not depend on ℎ, 𝑘, 𝑥. If, in addition, 𝛾(𝑡, 𝑥, 𝑣) = 𝛾(𝑡, 𝑥) (i.e., in the additive noise case), then (𝐸(¯ 𝑣(𝑡𝑘 , 𝑥) − 𝑣(𝑡𝑘 , 𝑥))2 )1/2 ≤ 𝐾ℎ.

(6.30)

Proof. In the case 𝛼 = 0 and 𝑏 and 𝜎 independent of 𝑣, the remainder 𝑟 in (6.28) is equal to zero (see (6.25)). Then we get (6.31) 1 1 𝑅(𝑡𝑘 , 𝑥) = 𝑅(𝑡𝑘+1 , 𝑥+𝑏ℎ+𝜎ℎ1/2 )+ 𝑅(𝑡𝑘+1 , 𝑥+𝑏ℎ−𝜎ℎ1/2 )+Δ𝑓 ℎ+Δ𝛾Δ𝑘 𝑤 +𝜌, 2 2 where 𝜌 satisfies (6.6) in the case of general 𝛾 and (6.7) in the case 𝛾(𝑡, 𝑥, 𝑣) = 𝛾(𝑡, 𝑥). Let us square (6.31) and then take conditional expectation with respect to 𝑡 the 𝜎-algebra ℱ𝑇𝑘+1 . We get (6.32) 𝑡

𝐸(𝑅2 (𝑡𝑘 , 𝑥) ∣ ℱ𝑇𝑘+1 ) =

1 2 1 𝑅 (𝑡𝑘+1 , 𝑥 + 𝑏ℎ + 𝜎ℎ1/2 ) + 𝑅2 (𝑡𝑘+1 , 𝑥 + 𝑏ℎ − 𝜎ℎ1/2 ) 4 4

1 + 𝑅(𝑡𝑘+1 , 𝑥 + 𝑏ℎ + 𝜎ℎ1/2 )𝑅(𝑡𝑘+1 , 𝑥 + 𝑏ℎ − 𝜎ℎ1/2 ) 2 𝑡 +ℎ2 (Δ𝑓 )2 + ℎ(Δ𝛾)2 + 𝐸(𝜌2 ∣ ℱ𝑇𝑘+1 )

𝑡

+(𝑅(𝑡𝑘+1 , 𝑥 + 𝑏ℎ + 𝜎ℎ1/2 ) + 𝑅(𝑡𝑘+1 , 𝑥 + 𝑏ℎ − 𝜎ℎ1/2 ))(Δ𝑓 ℎ + 𝐸(𝜌 ∣ ℱ𝑇𝑘+1 )) 𝑡

𝑡

+2Δ𝑓 ℎ𝐸(𝜌 ∣ ℱ𝑇𝑘+1 ) + 2Δ𝛾𝐸(Δ𝑘 𝑤𝜌 ∣ ℱ𝑇𝑘+1 ). Using Lemma 6.1 and the inequalities (6.23), one can estimate the terms in (6.32). For instance, 𝑡

∣ Δ𝛾𝐸(Δ𝑘 𝑤𝜌 ∣ ℱ𝑇𝑘+1 ) ∣ ≤ 𝐾∣𝑅(𝑡𝑘+1 , 𝑥)∣ 𝐶(𝜔)ℎ2 ≤

1 2 2 1 𝐾 𝑅 (𝑡𝑘+1 , 𝑥)ℎ + 𝐶 2 (𝜔)ℎ3 . 2 2

Introduce the notation 𝑚(𝑡𝑘 ) := sup 𝐸𝑅2 (𝑡𝑘 , 𝑥). 𝑥

After taking expectations in (6.32), we easily obtain 𝐸𝑅2 (𝑡𝑘 , 𝑥) ≤ (1 + 𝐾ℎ)𝑚(𝑡𝑘+1 ) + 𝐸𝜌2 (𝑡𝑘+1 ) + 𝐾ℎ3 , where the constant 𝐾 does not depend on ℎ, 𝑘, 𝑥. Therefore 𝑚(𝑡𝑘 ) ≤ (1 + 𝐾ℎ)𝑚(𝑡𝑘+1 ) + 𝐸𝜌2 (𝑡𝑘+1 ) + 𝐾ℎ3 , whence (6.29) follows because of 𝐸𝜌2 ≤ 𝐾ℎ2 in the multiplicative noise case (i.e., when 𝛾 depends on 𝑣) and (6.30) follows because of 𝐸𝜌2 ≤ 𝐾ℎ3 in the additive noise case. □ Introduce the equidistant space discretization as in (5.5). Using linear interpolation, we construct the following algorithm on the basis of the layer method (6.16):

2102

(6.33)


𝑣¯(𝑡𝑁 , 𝑥) = 𝜑(𝑥), ( 1 𝑣¯(𝑡𝑘 , 𝑥𝑗 ) = 𝑣¯ 𝑡𝑘+1 , 𝑥𝑗 + ℎ𝑏(𝑡𝑘 , 𝑥𝑗 , 𝑣¯(𝑡𝑘+1 , 𝑥𝑗 )) 2 + ℎ1/2 𝜎(𝑡𝑘 , 𝑥𝑗 , 𝑣¯(𝑡𝑘+1 , 𝑥𝑗 )) + 𝛼(𝑡𝑘 , 𝑥𝑗 , 𝑣¯(𝑡𝑘+1 , 𝑥𝑗 ))Δ𝑘 𝑤

1 ( + 𝑣¯ 𝑡𝑘+1 , 𝑥𝑗 + ℎ𝑏(𝑡𝑘 , 𝑥𝑗 , 𝑣¯(𝑡𝑘+1 , 𝑥𝑗 )) 2

− ℎ1/2 𝜎(𝑡𝑘 , 𝑥𝑗 , 𝑣¯(𝑡𝑘+1 , 𝑥𝑗 )) + 𝛼(𝑡𝑘 , 𝑥𝑗 , 𝑣¯(𝑡𝑘+1 , 𝑥𝑗 ))Δ𝑘 𝑤

)

)

+ℎ𝑓 (𝑡𝑘 , 𝑥𝑗 , 𝑣¯(𝑡𝑘+1 , 𝑥𝑗 )) + 𝛾(𝑡𝑘 , 𝑥𝑗 , 𝑣¯(𝑡𝑘+1 , 𝑥𝑗 ))Δ𝑘 𝑤, 𝑗 = 0, ±1, ±2, . . . , 𝑥𝑗+1 − 𝑥 𝑥 − 𝑥𝑗 𝑣¯(𝑡𝑘 , 𝑥) = (6.34) 𝑣¯(𝑡𝑘 , 𝑥𝑗 ) + 𝑣¯(𝑡𝑘 , 𝑥𝑗+1 ), 𝑥𝑗 ≤ 𝑥 ≤ 𝑥𝑗+1 , ℎ𝑥 ℎ𝑥 𝑘 = 𝑁 − 1, . . . , 1, 0. The following convergence theorem is proved for this algorithm using ideas from the above proof of Theorem 6.1 and also from the proof of an analogous theorem for an algorithm in the deterministic case [23, Chapter 7]. Theorem 6.2. Let Assumptions 6.1 hold. Let 𝛼 = 0 and 𝑏 and 𝜎 be independent of 𝑣. Then the error of the one-step approximation (6.33)-(6.34) with ℎ𝑥 = ϰℎ3/4 , ϰ > 0, is estimated as (6.35)

(𝐸(¯ 𝑣(𝑡𝑘 , 𝑥) − 𝑣(𝑡𝑘 , 𝑥))2 )1/2 ≤ 𝐾ℎ1/2 ,

where 𝐾 does not depend on ℎ, 𝑘, 𝑥. If, in addition, 𝛾(𝑡, 𝑥, 𝑣) = 𝛾(𝑡, 𝑥) (i.e., in the additive noise case) and ℎ𝑥 = ϰℎ, ϰ > 0, then (6.36)

(𝐸(¯ 𝑣(𝑡𝑘 , 𝑥) − 𝑣(𝑡𝑘 , 𝑥))2 )1/2 ≤ 𝐾ℎ.

We note that Remark 5.2 is valid here. 7. Numerical experiments 7.1. Model problem: Ornstein-Uhlenbeck equation. We consider the problem [ 2 2 ] 𝑎 ∂ 𝑣 ∂𝑣 ∂𝑣 −𝑑𝑣 = ∗ 𝑑𝑤(𝑡), (𝑡, 𝑥) ∈ [𝑇0 , 𝑇 ) × R, (7.1) + 𝑏𝑥 𝑑𝑡 + 𝛼 2 2 ∂𝑥 ∂𝑥 ∂𝑥 (7.2) 𝑣(𝑇, 𝑥) = 𝜑(𝑥), 𝑥 ∈ R, where 𝑤(𝑡) is a standard scalar Wiener process, 𝑎, 𝑏, 𝛼 are constants and (7.3)

𝜎 2 = 𝑎2 − 𝛼2 ≥ 0.

The solution of the problem is given by (see (2.6)-(2.7)): (7.4)

𝑣(𝑡, 𝑥) = 𝐸 𝑤 𝜑(𝑋𝑡,𝑥 (𝑇 )),

(7.5)

𝑑𝑋 = 𝑏𝑋𝑑𝑠 + 𝜎𝑑𝑊 (𝑠) + 𝛼𝑑𝑤(𝑠).

We have (7.6)

𝑋𝑡,𝑥 (𝑇 ) = 𝑒𝑏(𝑇 −𝑡) 𝑥 +

∫ 𝑡

𝑇

𝑒𝑏(𝑇 −𝜗) [𝜎𝑑𝑊 (𝜗) + 𝛼𝑑𝑤(𝜗)] .


2103

Clearly, the conditional distribution of 𝑋𝑡,𝑥 (𝑇 ) under 𝑤(𝑠), 𝑡 ≤ 𝑠 ≤ 𝑇, is Gaussian with the parameters (7.7)

𝑚(𝑡, 𝑥) := 𝐸 (𝑋𝑡,𝑥 (𝑇 )/𝑤(𝑠) − 𝑤(𝑡), 𝑡 ≤ 𝑠 ≤ 𝑇 ) ∫ 𝑇 𝑏(𝑇 −𝑡) 𝑥+ 𝑒𝑏(𝑇 −𝜗) 𝛼𝑑𝑤(𝜗) =𝑒 𝑡

𝑏(𝑇 −𝑡)

=𝑒

∫

𝑥 + 𝛼 (𝑤(𝑇 ) − 𝑤(𝑡)) + 𝛼𝑏

𝑡

𝑇

𝑒𝑏(𝑇 −𝜗) (𝑤(𝜗) − 𝑤(𝑡)) 𝑑𝜗,

𝛿 2 (𝑡) := 𝑉 𝑎𝑟 (𝑋𝑡,𝑥 (𝑇 )/𝑤(𝑠) − 𝑤(𝑡), 𝑡 ≤ 𝑠 ≤ 𝑇 ) ) 𝜎 2 ( 2𝑏(𝑇 −𝑡) 𝑒 = −1 . 2𝑏 From here, we get the following explicit solution of the Ornstein-Uhlenbeck equation: ( ) ∫ ∞ 2 1 (𝜉 − 𝑚(𝑡, 𝑥)) (7.9) 𝑣(𝑡, 𝑥) = 𝐸 𝑤 𝜑(𝑋𝑡,𝑥 (𝑇 )) = √ 𝜑(𝜉) exp − 𝑑𝜉. 2𝛿 2 (𝑡) 2𝜋𝛿(𝑡) −∞ (7.8)

We have (7.10)

𝑒𝑏(𝑇 −𝑡) ∂𝑣 (𝑡, 𝑥) = √ ∂𝑥 2𝜋𝛿 3 (𝑡)

∫

(

∞

2

(𝜉 − 𝑚(𝑡, 𝑥)) 𝜑(𝜉) [𝜉 − 𝑚(𝑡, 𝑥)] exp − 2𝛿 2 (𝑡) −∞

) 𝑑𝜉.

Now consider the problem which is a perturbation of (7.1)-(7.2): (7.11)

[

] 𝑎2 ∂ 2 𝑢 ∂𝑢 ∂𝑢 −𝑑𝑢 = 𝑑𝑡 + 𝛼 ∗ 𝑑𝑤(𝑡), (𝑡, 𝑥) ∈ [𝑇0 , 𝑇 ) × R, + (𝑏𝑥 + 𝜀𝑏1 (𝑥)) 2 2 ∂𝑥 ∂𝑥 ∂𝑥 (7.12) 𝑢(𝑇, 𝑥) = 𝜑(𝑥), 𝑥 ∈ R. Aiming to reduce variance in the Monte Carlo procedure, let us use the representation (4.2)-(4.3) with 𝜇 = 0 (hence 𝑌 ≡ 1) and with (7.13)

𝜆(𝑠, 𝑥) = −𝜎

∂𝑣 (𝑠, 𝑥), ∂𝑥

where ∂𝑣/∂𝑥 is from (7.10). We have (7.14)

𝑢(𝑡, 𝑥) = 𝐸 𝑤 [𝜑(𝑋𝑡,𝑥 (𝑇 )) + 𝑍𝑡,𝑥,1,0 (𝑇 )] ,

where 𝑋, 𝑍 satisfy the system (7.15)

𝑑𝑋 = [𝑏𝑋 + 𝜀𝑏1 (𝑋)] 𝑑𝑠 + 𝜎𝑑𝑊 (𝑠) + 𝛼𝑑𝑤(𝑠), 𝑑𝑍 = −𝜎

∂𝑣 (𝑠, 𝑋)𝑑𝑊 (𝑠). ∂𝑥

To realize (7.14), we use both the mean-square and the weak Euler procedures for solving (7.15) with respect to the Wiener process 𝑊 (𝑡). Due to Theorem 4.1, the variance is expected to be small if 𝜀 is relatively small. Here we exploit the control variates method (we put 𝜇 = 0) for variance reduction. Since in the SPDEs (7.1)(7.2) and (7.11)-(7.12) the coefficient 𝛼 is constant and 𝛽 = 𝛾 = 0, the method of important sampling can also be used without any theoretical difficulties (see the corresponding comment at the end of Section 4).

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7.2. Numerical results. For definiteness, we simulate (7.1)-(7.2) with 𝜑(𝑥) = 𝑥2 .

(7.16)

In the tests we fix a trajectory 𝑤(𝑡), 0 ≤ 𝑡 ≤ 𝑇, which is obtained with a small time step equal to 0.0001. To evaluate the exact solution given by (7.9), we simulate the integral in (7.7) by the trapezoidal rule with the step 0.0001. In Table 1 we present the results of simulating the solution of (7.1)-(7.2), (7.16) by the weak Euler-type scheme (3.7), (3.9). One can observe convergence with order one that is in good agreement with our theoretical results (see Remark 3.1 and note that in this example 𝛼 and 𝜎 are constant). In the table the “±” reflects the Monte Carlo error only, it gives the confidence interval for the corresponding value with probability 0.95. Similar results are obtained by the mean-square Euler scheme (3.2). For instance, for ℎ = 0.1 (the other parameters are the same as in Table 1) we get 𝑣ˆ(0, 0) = 0.7550 ± 0.0019. Table 1. Ornstein-Uhlenbeck equation. Evaluation of 𝑣(𝑡, 𝑥) from (7.1)-(7.2), (7.16) with various time steps ℎ. Here 𝑎 = 1, 𝑏 = −1, 𝛼 = 0.5, and 𝑇 = 10. The expectations are computed by the Monte Carlo technique simulating 𝑀 = 106 independent realizations. The “±”reflects the Monte Carlo error only, it does not reflect the error of the method. All simulations are done along the same sample path 𝑤(𝑡). The corresponding reference value is 0.73726, which is found due to (7.9). ℎ 0.2 0.1 0.05 0.02 0.01

𝑣ˆ(0, 0) 0.7735 ± 0.0019 0.7557 ± 0.0018 0.7456 ± 0.0018 0.7412 ± 0.0018 0.7391 ± 0.0018

To demonstrate the variance reduction technique from Section 4, we first simulate (7.1)-(7.2), (7.16) using a probabilistic representation of the form (7.14), (7.15) with 𝜀 = 0. In particular, we obtain that for 𝑀 = 104 , ℎ = 0.1 (the other parameters are the same as in Table 1) 𝑣ˆ(0, 0) = 0.7562 ± 0.0012 and for 𝑀 = 100, ℎ = 0.01 the approximate value 𝑣ˆ(0, 0) = 0.7383 ± 0.0010. Recalling that the results in Table 1 are obtained with 𝑀 = 106 Monte Carlo runs, one can see that we reach the same level of the Monte Carlo error with significantly fewer Monte Carlo runs. We note that although we use the optimal 𝜆(𝑠, 𝑥) here, the variance (and, consequently, the Monte Carlo error) is not zero, which is due to the error of numerical integration of the equation for 𝑍 in (7.15). Now we evaluate the solution of the perturbed Ornstein-Uhlenbeck equation (7.11)-(7.12). We take 𝜑(𝑥) from (7.16) and we choose a small 𝜀 > 0 and (7.17)

𝑏1 (𝑥) = −𝑥3 .

We simulate (7.14), (7.15), (7.16), (7.17) both without employing the variance reduction technique (i.e., we put 𝑍(𝑇 ) = 0 in (7.14)) and with variance reduction (i.e., using 𝜆(𝑠, 𝑥) from (7.13), (7.10)) by the Euler-type scheme (3.7), (3.9). The results of the experiments are presented in Table 2. When the variance reduction


2105

Table 2. Perturbed Ornstein-Uhlenbeck equation. Evaluation of 𝑢(𝑡, 𝑥) from (7.11)-(7.12), (7.16), (7.17) at (0, 0) with various time steps ℎ. Here 𝑎 = 1, 𝑏 = −1, 𝛼 = 0.5, 𝜀 = 0.1, and 𝑇 = 10. The expectations are computed by the Monte Carlo technique simulating 𝑀 = 104 independent realizations. The corresponding reference value is 0.6006 ± 0.0004, which is found by simulation with ℎ = 0.001 and 𝑀 = 106 . ℎ 0.2 0.1 0.02

without variance reduction 0.614 ± 0.014 0.611 ± 0.014 0.604 ± 0.014

with variance reduction 0.6014 ± 0.0042 0.6037 ± 0.0040 0.6001 ± 0.0041

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